text:
Business energy solutions
Find out how solar energy can reduce your business electricity costs.Date: January 6, 2017 10:34 AM File Name: 1609652 Small Business Energy Solutions FAQ Additional Details: Trim: 8.5x11 Finish Size: 8.5x11 Stock.
Business Strategy of Energy Solutions - Fuji Electric
Energy Saving Solutions Franchise | SuperGreen Solutions
Our EnergyRight programs let your company make smart energy choices while enhancing your bottom line.Energy Efficiency and Renewable Energy Programs, Solutions, and Funding Resources for Business.
We have helped all sizes of businesses reduce their gas and electricity costs.From small businesses to commercial and industrial organizations, Dominion Energy Solutions can provide business.
Columbia Gas of Ohio Energy SolutionsFrom easy-to-swap equipment upgrades to fully integrated energy plans, our solutions can lower utility bills and help you achieve better results with everything from.Our plans can help you save on electricity costs and use cleaner energy sources.
My Home Energy Right Solutions for the Home e-Score Home Improvement.We take care only to work with clients when energy solutions MAKE SENSE. read more.Entergy Arkansas offers a range of support to help improve your.This periodical newsletter will link you to practical and relevant solutions that will help your business use energy more efficiently.Perth Energy offer tailored solutions for businesses with transparent pricing and and annual tariff reviews for the best energy deals.Find companies using the Credibility Review Business Directory at DandB.com.It gives participating small to mid-size business customers the following: A no-cost energy audit that delivers a.
EnergySolutions is an international company headquartered in Salt Lake City.
Commercial Energy Experts | EC Business SolutionsBut with our money-back incentives it pays to become strategic about your energy use.We offer renewable power solutions that are reliable and customized to fit your needs.Our Small Business Energy Savings (SBES) offer provides a qualified ComEd Energy Efficiency Trade Ally energy expert to fully manage.
SuperGreen Solutions offers what no other business can: A thriving franchise model that takes advantage of large.View up to date company information for Energy Solutions and Services, Inc.Powering your business with reliable and economic solar energy direct from the sun means you can reduce your energy bills, protect yourself from rising electricity costs.


text:
February 2016 - 3 Night Bahamas (Port Canaveral Roundtrip) Cruise on Carnival Victory
Great Cruise
It was our first cruise and had a great time. Very little hidden fee's. Staff did a great job at keeping up with the crowed. Poor job at wiping down table's after picking up dirty dishes. Food was some of the best. The ship was amazing. The steward was the best at tending to our room. Carnival did a great job and after being married for 23 years this will be remembered as the perfect vacation. It is a must take cruise!!! For first timers they took good care of us.

subset: commoncrawl


text:
yamaha sz r new model 2015
Gallery Of Bajaj Discover 150 Vs Yamaha SZ RR Vs Honda CB UnicornYamaha SZ R 150cc Bike Launched New 2011 Yamaha SZR 150 BikeYamaha Motor Malaysia August 2015 New Model The Yamaha SZ16 is a light motorcycle manufactured by the India Yamaha Motor. Introduced in 2010, It comes in naked as well as faired variants. It has a carbureted single-cylinder, air-cooled, four-stroke engine, displacing 153 cc (9.3 cu in). A new model for the Philippine market was launched for 2015. Yamaha SZR660 is a sport bike with a 659 cc single-cylinder engine, produced by Yamaha Motor Corporation from 1996 to 2001.Yamaha SZ R 150 2014 al 2015 video versin yamaha sz rr version 2.0 new model 2015 - YouTube. 2015 Yamaha SZ-R 150 Caracteristicas Precio Colombia Walkaround.2013 Yamaha SZ 16 Walk Around. Download Fast Download. Yamaha SZR 70,000 kms Honest Ownership Review. View and download 2015 Yamaha SZ R 150 Caracteristicas Precio Colombia Walkaround in HD Video or Audio for free.Yamaha fz16 version2.0 New Model. Disclaimer DMCA. New Bikes.Discontinued. Summary. Yamaha SZ-R was a model sold in India before the SZ-RR version 2.0.Battery Type. Maintenance Free.
Yamaha SZ-R Review. 10 Dec, 2015 11:07 AM. Overview. This feature is not available right now.
Please try again later. Published on Aug 5, 2015. New Yamaha SZ-RR Version 2.0 2015 image compilation.Yamaha FZ S FI Version 2.0 ( V 2.0 ) Shark White 2016 - 2017 New Color Tone Model | India - Duration: 2:04. vamsi naik 196,515 views. Yamaha SZR SZ-R SZ R Modified Into 899 1199 Ducati Panigale.2015 Yamaha SZ16 Walkaround. A street model made for adventure and leisure, the Yamaha SZ can deliver with leading performance that can cater to everyday commuting needs. , 2015 Yamaha SZ-R 150 Caracteristicas Precio Colombia Walkaround.Here the all new yamaha sz rr 2017 model with bs4 engine. Yamaha SZR 2015. For sale by bijan mazumder 20 Feb 1:55 amAnderkilla, Chittagong. Tk 125,000. Negotiable. Yamaha SZR 2015.Model name: SZR. Condition: Used. Year: 2015. Model: SZ-R 150. Transmission: Manual. Mileage: 10,001 to 20,000 Kil. Fuel: Gasoline. Motorcyclemake: Yamaha.Login. Forgot your password? Register a new account. Contact via P24. Name Email Contact Number Message. 2015 Yamaha SZ-R 150 Caracteristicas Precio Colombia Walkaround.Play and Listen the new model of the yamaha r15 was recently unveiled in indonesia video includes detailed walkaround and the new 2017 yamaha yzf r6 heres everything you need to know about the 2017 yamaha 2017 View latest Yamaha SZ brochures online in Philippines. Download Free Brochure to know SZ motorcycle details, features, dimensions, mileage, styling and more.Yamaha SZR/ SZ16 loud pipe. Nov 19, 2015. Brand New Motorcycle Price In Bangladesh In 2018.Yamaha Sz R 150 2014 Al 2015 Video Versi 243 N Colombia Youtube. Class: sport. Production: 1995-1997. Related: Yamaha XTZ660 Tenere. Yamaha SZR660: specs. Yamaha SZR660: images, gallery. Yamaha SZR660: video. Yamaha SZR660: manuals, parts, microfiches. Yamaha SZR 660: Service Manual, 1995. The SZ, SZ-X and SZ-R are the finest range from this brand. To throw more light on the models, the SZ is theThe Yamaha SZ-R is priced at INR 56,000 (Ex-showroom, New Delhi) while the Yamaha SZ-X is priced atSuzuki Gixxer SF Review. Shiraz - June 15, 2015. Yamaha Saluto: Road Test Review. Get the latest Yamaha motorcycle model and enjoy a riding adventure like no other. Find New Yamaha SZ-RR Prices Mileage Photos Reviews Specifications Features Variants in India. Yamaha SZ16 price list for sale in Philippines 2015. The Yamaha Tracer 900 (FJ-09 in North America MT-09 Tracer in Japan, South America, Australia, and New Zealand) is a sport touring motorcycle first offered in 2015. The 3-cylinder crossplane engine comes from the MT-09 (FZ-09 in North America). My friend bought it of 18k so I planned to get RX but again I was stopped because my dad s condition was to get a brand new model.That was the 1st moment I saw Yamahas SZ, SZ-X model. Introducing the new 2015 Yamaha Motorcycles.Concept 2015 Yamaha 01GEN New Model. Behind the Scenes Industry News, Rumors, Spy Shots and information Manufacturer websites worldwide. New 2015 Yamaha YZF-R1 at EICMA [Live Photos]. Well aware that modern bikes are loaded with a lot of electronic amenities which optimize performance in Yamaha SZ RR V2.0.Sep 11, 2015 04:04 PM. Someone at the launch was saying that it was a late entry by Yamaha in the entry-level performance segment with its R3, but is it really true?These consist of 0 Yamaha upcoming bikes and 18 new Yamaha bikes. The list of Yamaha bike models in YAMAHA Models History. 70 in production. 58 discontinued.In 1978, Yamaha opened a new RD center in Minnesota, an office which could help the company expand its market coverage in the United States.2015 - present.YAMAHA SZR. 2 generations. Check out mileage, colors, images, specs, reviews of Yamaha SZ-RR at BikeDekho.cityname":"Delhi","popularity":83427,"modellaunchdate":"2015 -07-09T00:00:00Z"Stay ahead from all on Yamahas new bike. Yamaha SZ R Final TVC. 2015 Yamaha SZ-R 150 Caracteristicas Precio Colombia Walkaround.Yamaha SZ RR is the smooth and decent bike riding option for persons about 5.6 feet height. Music : Hear Break vibe tracks. Here the all new yamaha sz rr 2017 model with bs4 engine. 2015 Yamaha SZ-R 150 Caracteristicas Precio Colombia Walkaround.The new model based on Blue Core development ideal is powered by a newly designed air-cooled 149cc 4-stroke, SOHC, single-cylinder engine. See the Latest Yamaha sz r price in Bangladesh.Full specification,color,Design,feature,ownership review, experts comment, showroom address, tips for yamaha,yamaha szr 2018,yamaha sz r 2017 pricePrice of YAMAHA SZR Motorcycle in Bangladesh. Model : SZR ( New version With kick). Selling Reson Just Buy a New Model. Yamaha SZ-R 2015. 15 days, Khulna, Motorbikes Scooters. Brand: Yamaha. Model: Others. Year: 2015. KM Driven: 26,000 km.Keep your phone number private. Get the new OLX app instantly chat with the seller. Description, history and facts about Yamaha SZR. List tags and cars of Model Yamaha SZR.Yamaha SZR660 is a sport bike with a 659 cc single-cylinder engine, produced by Yamaha Motor Corporation from 1996 to 2001 Yamaha has a total of 18 models. The top 4 Yamaha models are- Yamaha YZF R15 V3The new Yamaha Fascino, which was unveiled at the 2018 Auto Expo, will be launched in India in March.I bought this bike the yamaha fzs v 2.
O in 2015. I have almost ridden 25000 km, for my daily SZ-RR Ver 2.0.India Yamaha Motor (IYM) Pvt. Ltd. today introduced the popular sports model, YZF-R3 at the Auto Expo 2018 in the presence of Bollywood star and Yamahas brand ambassador John Abraham.The new super sports model offers a 155 cc, liquid-cooled, 4-stroke, SOHC, 4-valve 2015 Yamaha SZ16 Walkaround. A street model made for adventure and leisure, the Yamaha SZ can deliver with leading performance that can cater to everyday commutingHere the all new yamaha sz rr 2017 model with bs4 engine.Music Credit :Giraffe Squad - Wait For Me [NCS Release][NCS] http Here the all new yamaha sz rr 2017 model with bs4 engine.This 2015, Yamaha Motor Philippines unveiled a new SZ at their Yamaha International Motorshow: The Motion of Innovation Part III held in Davao last August. Yamaha SZ R 150 2014 al 2015 video versin Colombia.Here the all new yamaha sz rr 2017 model with bs4 engine. Incolmotos Yamaha - Comodidad SZ-R. Rodada a pereira 2016. Review yamaha sz rr 150 blue core 2.0 - 2018. 2015 Yamaha SZ16 Walkaround. The Yamaha 2015 Bike Models DLC lets you expand your bike collection to include the eagerly anticipated YZF- R1, YZF-R1M, and YZF-R6 models! New tyrnew btryModel: SZ-R. Engine capacity: 155 cc.Yamaha FZ-S 2015. MEMBER32 days, Kurunegala, Motorbikes Scooters. BRAND NEW! Yamaha SZR660. What a chance to own one of these modern classics! PriceYamaha YAMAHA YFZ450R INJECTION BRAND NEW 2015 model. Yamaha SZ RR version 2.0 2017 Walk Around Review Special Edition Matt Green New01:10. 2015 Yamaha SZ-R 150 Caracteristicas Precio Colombia Walkaround.2.0 yamaha sz rr top speed sz rr new model sz rr 2016 yamaha bikes yamaha bikes 2016 yamaha sz rr latestBikesDinos: Yamaha SZ RR Version 2.0 Walkthrough (price, mileage, etc.) 7 January, 2015.2015 Yamaha SZ-R 150 Caracteristicas Precio Colombia Walkaround. 18 December, 2014. This is my Yamaha SZR/SZ16 modfied stainless muffler.2015 Yamaha SZ16 Walkaround. A street model made for adventure and leisure, the Yamaha SZ can deliver with leading performance that can cater to everyday commuting needs. Yamaha SZ-R OwnersFans updated their cover photo. June 1, 2015 .Yamaha SZ-R OwnersFans added 2 new photos.SZ-X and SZ-R both are same, only disc brake is added in SZ-R model. New Cars.Yamaha SZ-R Mileage. This Yamaha motorcycle is truly a fabulous bike model of this time. Offering 55 kmpl mileage in city and 64 kmpl mileage on highway, it has become an all-time favorite motorcycle of bikers. Yamaha SZR660 short ride, Swann Freestyle camera.Yamaha SZR 660 exhaust sound, acceleration and fly by compilation - Duration: 0:49. Yamaha SZ-R смотреть онлайн | Бесплатное видео в HD качестве без рекламы, без смс и без регистрации View the best Yamaha SZR 660 (szr660) motorcycle pictures uploaded by users all over the world.Add a New Motorcycle. Manufacturer. Year. Model. Add Motorcycle. Yamaha sz16 Motortrade philippines motorcycle dealer yamaha, get latest yamaha motorcycle model enjoy riding adventure choose budget ll match perfect motorcyle.81 sz 16 yamaha motorcyle. Underbone tuner modified. Yamaha kicksoff 2015 grand. Yamaha Sz Rr Version 2.0 Review Special Edition Matt Green New Model 2017 India.2015 Yamaha Sz R 150 Caracteristicas Precio Colombia Walkaround. F. Sz R 150cc Yamaha Modified By Sherwin. Nueva yamaha sz-rr 2016 espaol precio colombia. 22 December 2015.New yamaha sz rr version 2 0 bs IV aho walk around detail review.
related notes
- chase bank atm mexico city
- run for water discount code
- baixar convertxtodvd serial gratis
- assistir filme jogos vorazes 2015
- raspberry pi matchbox keyboard not in accessories


text:
|Click Here to Login
You may also like:
« Non GFCI outlets won't work on Sunseeker 2690 | What do you like and dislike about your Class A? »
Our communities encompass many different hobbies and interests, but each one is built on friendly, intelligent membership.
» More about our Communities
Our Marine websites focus on Cruising and Sailing Vessels, including forums and the largest cruising Wiki project on the web today.
» More about our Marine Communities


text:
Happy International Happiness Day! This is a preamble into an upcoming article on hacking happiness with actual numbers and charts and graphs.
And, as always, this is also an excuse to show off adorable dog pictures:
Continue reading
Happy International Happiness Day! This is a preamble into an upcoming article on hacking happiness with actual numbers and charts and graphs.
And, as always, this is also an excuse to show off adorable dog pictures:
Continue reading
Recently I've had the pleasure of corresponding with an absolutely lovely young couple that wanted some advice. For anonymity's sake, let's call them…Amelia and George. Amelia and George are wonderful savers, partially due to necessity right now, as George is a full time student and Amelia is the current breadwinner with a small salary at a nonprofit. They both have sizeable side-hustles that are bulking up their income, but money is tight.
And they really want a house, possibly even a tiny house. Oh, and they have no credit score to speak of. Oh, and they may have to move in two years after George has graduated. Oh, plus, since we first started communicating months ago, a decision has already been made – the suspense!
In this post, we'll cover what SSL Certification is (short answer: that "https" thing), why bloggers should care about it (short answer: don't freak out if you don't have it), and how to get it entirely free (Fergus did it in roughly half an hour!).
Steve from Think, Save, Retire wrote a great piece on what SSL is, why it matters, and why it generally does not matter much to blogs. Steve's got a wonderful blog, has very recently retired, and is super supportive in the Personal Finance blogging community.
There are, however, two additional things about SSL certification that are important to note that prompted us to write this article:
It's not like we have never donated to charities before. We've sent the Red Cross the odd donation, and we've donated on behalf of coworkers running for charity. We've even pitched in for a couple Go Fund Me's for friends in tough situations. It was just never a lot – $200-$300 a year at the most.
This year, we're 10Xing that.
Last Saturday there was a ten alarm fire in Cambridge, MA, the largest fire in the area since the 80s. Thankfully, no one was seriously injured, but still – holy shit. Eleven buildings, over 60 people affected, multiple fire departments on the scene, and now that area looks unrecognizable.
It started scarily close to Fergus's grad school apartment, making me even more thankful nothing similar happened to that fire trap of a house he lived in for years. Seriously – an electrician came by once to fix a problem and basically said "This house would be up in flames instantly if there was a fire," which unfortunately is not that uncommon with the older homes in Cambridge. And of course those living in these fire traps are often lower income, students, or immigrants – people who don't have a choice.
We know at least one family that's been displaced. Apartment, car, all material possessions – gone during one awful day. And no rental insurance. Like many others, they took to crowd funding (we contributed), but not everyone can rely on that.
Rental Insurance covers a few different things, the core being (more detail below):
I just got a new computer. And I love it.
I primarily wanted something for writing (this blog + other random projects), and for general checking email, browsing the web and the like. I needed something lightweight and portable, something I could pick up on a whim for hours of creation. I went with a Chromebook and am very happy with my decision.
And you, too, can be just like me!
In all seriousness, that is my actual earning statement after officially blogging for 13 months (well, more like -$99 due to hosting and domain name). I'm writing this post for two reasons:
(1) It is insanely easy to glamorize the lives of your favorite bloggers, or to envy those that actually make a living at this; they make it look so easy. When you think, "Hey, I have ideas, I want to blog, too," it's disheartening to see what appears to be everyone else, except for you, succeed.
This is called survivorship bias. Lots of people fail or struggle, you just don't see it. This post is me showing the world (okay, the optimistically nine people who regularly read this blog), that I have tried and failed at generating any kind of "side hustle" income from this blog. That I have spent easily over 100 hours of just writing for negative financial returns. We so rarely publicize and try to learn from failures.
The weekend before last we went to the Boston Local Food Festival on the Greenway in Boston. While we used the opportunity as our one fancy "nom out" for the month (see this post for details on our monthly adventures), there were loads of free samples, a cooking competition, and demonstrations all for free. We even got free parking using Spot Hero**).
There are so many great, free activities in any city, and this post is a guide to what's available in our own neck of the woods. However, most of the tips are valid across the country. This doesn't cover the most touristy things you can do for free in Boston (freedom trail, etc), but Boston on Budget has a great article covering that base for your reference.
This is not a sponsored post, but there are affiliate links to TransferWise, a company we enthusiastically support and reached out to.
I love technology. So much information at our fingertips, such almighty power compared to twenty years ago. So why is so much banking technology awful? At work I could spin up a network of VMs in minutes, so why does it take multiple days to make a transfer of funds from one bank in Massachusetts to another bank in Massachusetts? Why? Why does it need to be that way?
The worst culprit by far has been international transfers and exchanges. We have family outside of the US, and occasionally we like to send money overseas. Our first experience with transferring money was just painful. We searched around, trying to find the best combination of rate and fee for our use case.
These past 6 weeks have been crazy busy. Three work trips and a 10 day vacation totaling $5883 (over five months rent!!).
The work trips were actually pretty great in their own rights. I was able to get in some sightseeing, meet up with family I haven't seen since I was literally a baby, and grow friendships. I've always been fairly friendly with work colleagues, but now I'm building more honest-to-goodness friendships – the kind where we would actually keep in touch if I quit work tomorrow. It's pretty awesome.
And that vacation? Best ever, hands down. Me and Fergus, my 80-year-old grandparents, and two of their "retired" friends spent 10 days in the Cayman Islands full of sting rays, sunscreen slathering, and scuba diving. Yes, scuba diving. Continue reading


text:
Young Chop Says Chief Keef And Lil Durk's Beef Isn't Serious
Producer Young Chop recently sat down for an interview with VladTV and discussed the alleged tension between Chicago rappers Chief Keef and Lil Durk. The gregarious producer was quick to dismiss the alleged beef, saying "It's just a little brother, big brother situation." Though the disagreement between the two recently spilled over to Twitter, Chop doesn't think the situation is serious. Lil Durk discussed his issues with Chief Keef back in July during a radio interview, saying that he "had a couple of words" with Keef but "he wasn't sneak dissing." Watch Chop's interview above.
Young Chop recently announced that he'll drop his Precious project on September 12.
[via VladTV]


text:
E1: DB: How to Delete SQL Packages on the AS/400
Last updated on OCTOBER 04, 2017
Applies to:JD Edwards EnterpriseOne Tools - Version 8.98 and later
IBM i on POWER Systems
How do I delete SQL Packages (SQLPKG) on the AS/400?
Sign In with your My Oracle Support account
Don't have a My Oracle Support account? Click to get started
My Oracle Support provides customers with access to over a
Million Knowledge Articles and hundreds of Community platforms


text:
- Web Content
- Blog Posts
- News Articles
- Interviews
- Case Studies
Kristin Hanes can provide top-notch, SEO-ready content and blog posts for your website. Kristin Hanes has been published in outlets such as the San Francisco Chronicle's SF Gate, Stash Invest, Marie Claire, and The Bold Italic, among others. She's fast, deadline-driven, and knows how to write content that will appeal to a wide variety of audiences. Kristin's speciality is taking complex topics and breaking them down into engaging, interesting articles and blog posts.
- Press Releases
- Media Outreach
- Event Planning
- Spokesperson
KristinHanes.com can provide media outreach to help get your business in the news. We're a network of experienced journalists with connections up and down the West coast. Our staff members can serve as a media spokesperson on television, in print and on the radio. We have a track record of success.
- Social Media Posting and Scheduling
- Copy Writing for Social Media
KristinHanes.com can provide social media management for your company, including Instagram, Pinterest, Twitter and Facebook strategy. Kristin Hanes has a known track record of growing web traffic through Pinterest, as evidenced by her own popular blog, www.thewaywardhome.com.
- Corporate and instructional videos
- Audio news stories and features
- Video and audio editing
- Narration and voice-over
Why Choose Kristin
- Deadline-driven, fast copy and news writer
- Keen awareness of the news media and what grabs reporters
- Talented press release and pitch writer
- Fast audio and visual editing skills
- Optimistic, upbeat and personable
What Clients Say
Kristin was great at landing coverage for my business, Babyworks. Her press releases and media outreach got me interviews in television, radio and newspapers in Portland. I'd highly recommend her media services.


text:
Asymptotic normality of the Nadaraya-Watson estimator for non-stationary functional data and applications to telecommunications.
- Dwayne Lynch
- 2 years ago
- Views:
1 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary fuctioal data ad applicatios to telecommuicatios. L. ASPIROT, K. BERTIN, G. PERERA Departameto de Estadística, CIMFAV, Uiversidad de Valparaíso, Chile Istituto de Matemática y Estadística Rafael Laguardia, Facultad de Igeiería, Uruguay. July 6, 2007 Abstract We study a o-parametric regressio model where the explaatory variable is a ostatioary depedet fuctioal data ad the respose variable is scalar. Supposig that the explaatory variable is a o-statioary mixture of statioary processes ad geeral coditios of depedece of the observatios implied i particular by weakly depedece, we obtai the asymptotic ormality of the Nadaraya-Watso estimator. Uder some additioal regularity assumptios o the regressio fuctio, it ca be obtaied asymptotic cofidece itervals for the regressio fuctio. We apply this result to the estimatio of the quality of service o the Iteret where the cross traffic is o-statioary. Keywords: No-parametric regressio, fuctioal data, asymptotic ormality, o-statioarity, depedece, quality of service. AMS subject classificatio: 62G05, 62G08, 62G20. Itroductio We study the problem of estimatig a o-parametric regressio fuctio φ whe the explaatory variable is fuctioal ad may be o-statioary ad depedet. We observe {X, Y } = {X i, Y i : i = 0,..., } such that Y i = φx i + ε i, i = 0,...,, where φ is a fuctio φ : D R, X i D, Y i R, the ε i s are cetered ad idepedet of the X i s ad D is a semi-ormed liear space with semi-orm. Istituto de Matemática y Estadística Rafael Laguardia, IMERL, Facultad de Igeiería, Julio Herrera y Reissig 565, Motevideo 200, Tel-fax: , s: Departameto de Estadística, CIMFAV, Uiversidad de Valparaíso, Gra Bretaña 09, 4to piso, Playa Acha, Valparaíso, Chile, Tel: ,
2 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 2 Our aim is to fid the asymptotic distributio of the estimator i= φ Y x Xi ik h x = 2 i= K x Xi h for x D where the kerel K is a positive fuctio ad the sequece h > 0 is the badwidth. We use the covetio 0/0 = 0. The estimator 2 is a geeralizatio of the Nadaraya-Watso estimator cf. []. The asymptotic ormality of the Nadaraya-Watso estimator is proved i [2] whe the observatios X i, Y i are idepedet, ad for depedet observatios, i [3]. Ferraty, Goia ad Vieu [4], [5] proposed the estimator 2 to estimate the regressio fuctio whe the explaatory variable is fuctioal. For weakly depedet ad statioary observatios, they proved the complete covergece of 2 ad obtaied rates of covergece whe the fuctio φ satisfies some regularity coditios. Aspirot et al [6] have geeralized the result of complete covergece for the estimator 2 to a o-statioary case. Masry 2005 [7] has proved the asymptotic ormality of φ for statioary weakly depedet radom variables. Ferraty, Mas ad Vieu 2007 [8] cosider asymptotic ormality of the estimator φ i the case of idepedece from theoretical ad practical poit of view. Related work o desity estimatio with fuctioal data ca be foud i [9] ad [0] for mode estimatio ad i [] for asymptotic ormality of kerel estimators. Surveys o fuctioal data aalysis ca be foud i Ramsay ad Silverma 2005 [2] ad i Ferraty ad Vieu 2006 [3] for the o-parametric case. Our goal i this article is to geeralize the result of [7] to the case whe X is a ostatioary mixture of statioary processes. More precisely, we suppose that there exist two radom processes ξ = {ξ : N}, Z = {Z : N} ad a fuctio ϕ : D R D such that ξ is statioary with values i D, ξ is idepedet of Z, Z takes values i a fiite set {z,...,z m } of R ad X satisfies X i = ϕξ i, Z i, i = 0,...,. 3 We suppose geeral coditios of depedece cf. assumptios i Subsectio 3., implied i particular by weakly depedece of the observatios ad of the process ξ, but the process Z may be o-statioary ad o-weakly depedet. I Theorem, uder these coditios ad a set of techical assumptios, we obtai the asymptotic ormality of the estimator 2. This result is based o cetral limit theorems for fields idexed i N or Z d. I Propositio ad 2, we prove results of cetral limit theorems for triagular arrays of a R m -valued cetered statioary radom field idexed i N or Z d. For these fields the existece of a cetral limit theorem depeds strogly o the geometry of the subset of Z d where the observatios are idexed. We apply these results to our model, i which we have a mixture of statioary fields X = ϕξ, Z where Z is o-statioary but X = ϕξ, Z coditioed to Z is statioary. We cosider here our observatios idexed i the level sets of Z. Some assumptios about the field Z should be made i order to esure properties of the level sets ad to have a cetral limit theorem i this case. I theorem 2, supposig additioally some regularity assumptios o the regressio fuctio φ ad o the badwidth h, we obtai the asymptotic ormality of φ x φx for x D, which allows us to costruct asymptotic cofidece iterval for φx.
3 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 3 The o-statioary assumptio for the data appears aturally whe modellig the Iteret traffic cf. [4], [5], [6]. I this paper, we are iterested i estimatig the quality of service for voice ad video applicatios. We call quality of service parameters some etwork characteristics like delay, packet loss, etc.. The iterest i voice or video applicatios is based i the fact that these etwork traffic the same as may real time applicatios have high quality of service requiremets very low delay ad losses, etc.. I Sectio 4, we use the model 3 to estimate quality of service parameters for traffic over the Iteret. We assume that these parameters are a fuctio of the Iteret traffic, represeted by X = ϕξ, Z. I the literature, it is usual to assume that the traffic is piecewise statioary, i this model Z is a o-statioary radom process that selects betwee differet statioary behaviors. The process Z idicates for example if the etwork is very much loaded or ot ad it could describe seasoal or periodic behaviors. The process ξ is related with obtaied measuremets variatios. We give simulatios of the estimator 2 ad cofidece iterval, for simulated ad real traffic o the Iteret. The orgaizatio of the paper is as follows. I Sectio 2, we give some prelimiaries o the otios of asymptotically measurable sets ad fields ad state a cetral limit theorem for triagular arrays of R m -valued cetered statioary radom fields. I Sectio 3, we give our result o the asymptotic ormality of the estimator φ ad we make the assumptios uder which we obtai this result. Sectio 4 describes the applicatio of our result to the problem of estimatig the quality of service o the Iteret. Sectio 5 is devoted to the proofs. 2 Prelimiaries I this sectio we preset some prelimiary results. I subsectio 2. we defie the otio of asymptotically measurable subsets that allows, i subsectio 2.2, to give a cetral limit theorem for triagular arrays of R m -valued cetered statioary radom fields. I subsectio 2.3 we preset the otio of asymptotically measurable field that allows to state a cetral limit theorem for a field described by equatio 3. I this sectio, we cosider both the case whe the field is idexed i N ad i Z d ad i the followig, L stads for N or Z d. Whe we have L = N, we deote l = ad for a set A N, A = A {0,,..., } ad A c = A {0,,..., }. Whe we have L = Z d, we deote l = 2 + d ad for a set A Z d, A = A {,...,} d ad A c = A c {,...,} d. 2. Notio of asymptotically measurable subsets I what follows carda idicates the cardial of subset A. Defiitio. A subset A L is said to be a asymptotically measurable subset if, for all k L, the followig limit exists Fk, A = lim card{a c A k}. l The fuctio F., A is called the border fuctio of the subset A. Defiitio 2. The subset family { A i : i =,...,m } i L is a asymptotically measurable family if, for all i {,...,m}, the subsets A i are asymptotically measurable ad, for all k L ad
4 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 4 i, j {,...,m}, the followig limit exists F k, A i, A j = lim card{a i A j k}. l 2.2 Cetral limit theorem for triagular arrays of R m -valued cetered statioary radom field I the followig, the otatio = w meas covergece i law ad N d 0, Σ represets a zero mea ormal distributio i dimesio d with covariace matrix Σ. Moreover, for X = Xk k L a R m -valued cetered statioary radom field, we deote the compoets of X by X = X,,...,X m, ad X k = X, k,...,xm, k for k L. Defiitio 3. We defie the class BL as the class of R m - valued cetered statioary radom fields X = X k k L that satisfies the followig coditios: H For all i, j {,...,m} ad N, we have { } E X i, 0 Xj, k <. k L H 2 Let X,J = X,,J,...,X m,,j be the trucatio by J of X, defied for k L ad J > 0 by [ ] X,J k = Xk { Xk J} E Xk { Xk J}, where represets the euclidia orm o R m. i There exists a sequece γk 0 such that k L γk < ad such that for all k L, N, i, j {,...,m} ad J > 0 we have E { X i,,j 0 X j,,j k } γk. ii There exists a sequece bj such that lim J bj = 0 ad for all set B L, i {,...,m}, N ad J > 0 where E [ S B, X i, S B, X i,,j ] 2 bjcardb, l S B, X i, = l /2 X i, k. k B H 3 There exists a sequece of reals umbers CJ such that for all B L, N, J > 0 ad i {,...,m} we have [ E S B, X i,,j ] 4 cardb 2 CJ. l
5 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 5 H 4 There exist a fuctio h : R + R +, such that lim x + hx = 0 ad a fuctio g : R + R m R +, with gj, t < for all fixed J > 0 ad sup t R m gj, t = g J <, such that [ ] [ ] [ ] E e isb C, t,x,j E e isb, t,x,j E e isc, t,x,j gj, thdb, C, for all disjoit sets B, C L, all N, J > 0 ad t R m. Here, represets the scalar product o R m. H 5 There exist sequeces γ J i, j, k ad γi, j, k, i, j {,...,m}, k L such that for all J > 0 we have { } lim E X i,,j 0 X j,,j k = γ J i, j, k, ad for i, j {,...,m} ad k L. We have the two followig propositios proved i Sectio 5. lim J γj i, j, k = γi, j, k 4 Propositio. If X = X k k N belogs to BN, the for ay asymptotically measurable family { A i : i =,...,m } i N, we have as teds to where for i, j {,...,m} S A, X,,...,S A m, X m, w = N m 0, Σ, Σi, j = γi, j,0fi, j,0 + k {γi, j, kfi, j, k + γj, i, kfj, i, k} ad Fi, j, k = lim card{a i A j k}. Propositio 2. If X = X k k Z d belogs to BZd, the for ay asymptotically measurable family { A i : i =,...,m } i Z d, we have as teds to S A, X,,...,S A m, X m, w = N m 0, Σ, where for i, j {,...,m} ad Σi, j = k L γi, j, kfi, j, k card{a i A j k} Fi, j, k = lim 2 + d.
6 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Notio of asymptotically measurable field I what follows a.s. meas almost sure covergece as teds to ad A is the idicator fuctio of a set A. I the case L = N, we set D = {0,..., } ad if L = Z d, we set D = {,...,} d. Defiitio 4. The R m -valued radom field Z = Z L is a asymptotically measurable field i L if there exists a radom probability measure R 0 i B, where B is the Borel σ-algebra i R m, such that a.s. R 0 B l m D {Zm B} ad for all k L {0} there exists a radom measure R k i B 2, where B 2 is the Borel σ-algebra i R 2m, such that for all B, C B l m D {Zm B} {Zm k C} a.s. R k B C. If the limit measure is o-radom Z is called a regular field i L. The followig propositio cf. Proof i [7] relates asymptotically measurable ad regular fields with asymptotically measurable subsets. Propositio 3. If Z = Z L is a asymptotically measurable field, B, B 2,...,B m are disjoit subsets of B ad A i = { L : Z B i } for i =,...,m, the, coditioally to Z the family {A,...,A m } is a asymptotically measurable family with Fk, A i, A j = R k B i, B j ad F0, A i, A j = R 0 B i δ ij, where δ i j is the Kroecker delta. Remarks: I the ext sectios we will assume further hypotheses o Z for the field described i 3 i order to prove the results, but the previous propositio is the tool for coditioig o level sets of the field Z ad applyig the results for statioary fields to the o-statioary case 3. 3 Results I the followig, we cosider a fixed x D ad we will study the asymptotic ormality of φ x give by 2. Proofs of the results are i Sectio 5. The estimator φ ca be also writte φ x = g x f x, where g x = f x = Y i K X i, ψh K X i 5 ψh
7 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 7 with u x K u = K, u D. 6 ad ψh is give i assumptio A. We recall that we use the covetio 0/0 = 0. h I Subsectio 3., we give the assumptios uder which the asymptotic ormality of φ x is obtaied ad i Subsectio 3.2 we give our results. 3. Assumptios We make the followig assumptios o the distributio of X, Y. A There exist positive fuctios ψ, ψ,...,ψ m defied o R + D, fuctios c,...,c k defied o D ad a subset of {,...,m} such that for all h > 0 P [ ϕξ, z k x h] = c k xψ k h, x, ψ k h, x where lim h 0 ψh, x = if k, ad lim ψ k h, x h 0 ψh, x = 0 if k C. I the followig, to simplify the otatio, we replace ψh, x by ψh, but the quatity ψh still depeds of x. A 2 The fuctios u ψ k u, x are differetiable o R +, with differetial fuctio deoted ψ k u, x ad satisfy lim h 0 h ψ k h, x where the d k s are fuctios defied o D. 0 Kuψ k uh, xdu = d kx, A 3 The process Z is regular ad satisfies for k {,...,m} as teds to that ψh {Zi =z k } P PZ i = z k = 0, 7 where P = meas covergece i probability. For k {,...,m}, deote by p k the limit p k = lim PZ i = z k. Let the R 2m -value radom process X = X,,..., X 2m, defied for i N as follows: for l {,...,m} X l, i = ψh K ϕξ i, z l φϕξ i, z l + ε i ψh E [K ϕξ i, z l φϕξ i, z l ],
8 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 8 for l {m +,...,2m} X l, i = where K is defied by 6. ψh K ϕξ i, z l Moreover we assume the followig hypotheses. A 4 The R 2m -value radom process X belogs to BN. A 5 The fuctio φ is cotiuous. ψh E [K ϕξ i, z l ], A 6 The estimator f x defied by 5 coverges i probability to fx > 0 where the fuctio f is defied by fu = k p k d k uc k u, u D. We suppose the followig hypothesis o the estimator φ. A 7 The fuctio K is a positive fuctio with support [0, ]. A 8 The badwidth h satisfies ad Remarks o the assumptios: lim h = 0 lim ψh =. The coditios A ad A 2 mea that, coditioally o Z, the distace betwee the observatio X i ad the estimatio poit x has a desity. The quatity ψ k h, x plays a role aalogous to h d for multidimesioal estimatio. The hypothesis A 3 meas that the variable Z does ot behave too irregularly. It is assumed that the process has some kid of statioarity i mea. However this is ot a strog assumptio, ad for example A 3 could be satisfied by periodic or semi-periodic radom variables. The relatio 7 is satisfied for example if the variables Zi =z k satisfy a cetral limit theorem for each k =,...,m. The hypothesis A 4 ca be replaced by various sets of hypothesis doig a trade-off betwee coditios o the momets of Y ad coditios of mixig. Perera 997 [8] describes several sets of coditios that imply the previous hypotheses. I the hypothesis A 6, fx > 0 meas roughly that there is a positive probability of fidig observatios X i ear i the sese of the semi-orm o D the poit x where we wat to compute the estimator. The fuctio f play the role of the desity of the whole process X = ϕξ, Z. Moreover the covergece i probability ca be obtaied uder the coditios A y A 2, coditios o the momets of Y i ad coditios of mixig cf. Proof i [6]. The hypothesis A 5, A 7 ad A 8 are classical for obtaiig asymptotic ormality.
9 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Asymptotic ormality of φ x The asymptotic ormality of φ x is obtaied i three stages: asymptotic ormality of the field X i Propositio 4, of g x Eg x, f x Ef x i Propositio 5 ad the of φ x i Theorem. Theorem 2 gives the asymptotic ormality of φ x φx which allows to costruct asymptotic cofidece iterval for φx. Propositio 4. For l {,...,m}, deote A l = {k N, Z k = z l } ad A l+m = A l. If Z is asymptotically measurable, the uder Assumptio A 4, coditioally o Z, we have, as teds to S A, X,,..., S A 2m, X 2m, w = N 2m 0, Σ, where for i, j {,...,2m} Σi, j = γi, j,0fi, j,0 + k N {γi, j, kfi, j, k + γj, i, kfj, i, k}, with γi, j, k, k N, defied by 4 for the field X ad Fi, j, k = lim card{a i A j k}. Propositio 5. Uder Assumptios A 3 ad A 4, we have, as teds to, that w ψh g x Eg x, f x Ef x = N 2 0, A, where with a x = A = Σi, j, a 2 x = i,j= a x a 2 x a 2 x a 3 x 2 i= j=m+ Σi, j, a 3 x = 2 i,j=m+ Σi, j. Theorem. Uder Assumptios A A 8, we have as teds to ψh φ x Eg x w = N 0, σ 2 x, Ef x where Theorem 2. We suppose that B the fuctio φ satisfies with β > 0 ad L a positive costat, σ 2 x = a x 2a 2 xφx + a 3 xφ 2 x f 2. 8 x φu φv L u v β,
10 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 0 B 2 there exist M > 0 ad a fuctio α such that for each k {,...,m} ad h > 0, we have P [ ϕξ, z k x h] c k x ψ k h, x Mαh α > 0. B 3 the badwidth h satisfies ad lim αh ψh = 0 lim hβ ψh = 0 B 4 for each k {,...,m}, we have lim ψh PZ i = z k p k = 0. The uder Assumptios A A 8, B B 4, we have as teds to, where σ 2 x is defied by 8. ψh φ x φx w = N 0, σ 2 x, 4 Applicatio to ed-to-ed quality of service estimatio 4. Motivatio ad model Nowadays ew services are offered over the Iteret like voice ad video. For these ew applicatios the eed to measure the etwork performace has icreased. Multimedia applicatios have several requiremets i terms of delay, losses ad other quality of service parameters. These costraits are stroger tha the oes for usual data trasfer applicatios mail, etc.. Measuremets of the Iteret performace are ecessary for differet reasos, for example to advace i uderstadig the behavior of the Iteret or to verify the quality of service assured to the ew services. There are several measurig techiques, most of them with software implemetatios, that ca be classified i active sedig cotrolled traffic called probe packets ad passive measures geerally measures at the routers. The aim of these techiques is to detect differet characteristics of the etwork that ca be the topology or some performace parameters delay, losses, available badwidth, etc.. There are also several problems to measure the performace parameters, for example the route of the packets ca chage, the traffic bit rate is ot costat ad ormally is i bursts, the probe packets ca be filtered or altered by oe ISP Iteret Service Provider i the path, there is ot clock sychroizatio betwee routers ad ed equipmets, etc.. Normally the iteral routers i the path betwee two poits of iterest are ot uder the cotrol of oly oe user or oe ISP. Therefore, it is ot very useful to have measurig procedures that deped o the iformatio of the iteral routers. For this reaso ed-to-ed measures is oe of the most developed methodologies durig the last years cf. [9], [20],[2]. We cosider a sigle lik ad a voice or video traffic that we wat to moitor, that is we wat to kow the quality of service parameters whe this traffic goes betwee two poits over
11 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data the etwork. The lik is shared by this multimedia traffic ad may other traffics i the etwork that are ukow ad that is called cross traffic ad could be modelled as a stochastic process. The performace parameter for packets of a multimedia traffic could be delay, packets losses, delay variatio or all of them together ad we will cosider it as a radom variable Y. This performace parameter could be modeled as a fuctio of the cross traffic stochastic process, the video or voice stochastic process, the lik capacity ad the buffer size. We wat to estimate Y without sedig video or voice traffic durig log time periods. Active measuremets, that sed traffic, are useful if they do ot charge the etwork, so they cosist o small probe packets set durig short time periods. The lik capacity ad the buffer size are ot kow but it is assumed that they are costats durig the moitorig process. The multimedia traffic is also kow therefore we ca suppose that Y = φx where X is a characterizatio of the cross traffic stochastic process. We ca ot measure the cross traffic process but we ca obtai data closely related with this process. This data is described i the followig subsectio. The cross traffic process o the Iteret is a depedet o-statioary process c.f [4] or [5]. To take ito accout o-statioary behaviors we will suppose that X satisfies the relatio Measuremet procedure I what follows we describe the procedure to estimate the fuctio φ. We divide the experimet i two phases: first, we sed a burst of small probe packets pp of fixed size spaced by a fixed time. Immediately after the burst we sed durig a short time a video stream. We repeat the previous procedure durig some time lapse, sedig a ew burst ad a video probe after a time iterval measured from the previous ed of the video stream as it is show i the scheme i figure. pp video Figure : Measuremet procedure scheme With the probe packets burst we ifer the cross traffic of the lik. We measure at the output of the lik the iterarrival time betwee cosecutive probe packets. This time serie is strogly correlated with the cross traffic process that shares the lik with the probe traffic. We compute the empirical distributio fuctio of probe packets iterarrival times because this distributio is related with the queue behavior i the etwork. Usig this cross traffic estimatio ad measurig the delay Y over the video, we wat to estimate the fuctio φ. From each probe packet burst ad video sequece j we have a pair X j, Y j, where X j is the empirical distributio fuctio of iterarrival times ad Y j is the performace metric of iterest measured from the video stream j. Therefore, our estimatio problem has bee trasformed ito the problem of iferrig a fuctio φ : D R where D is the space of the probability distributio fuctios ad R is the real lie such that the X j, Y j satisfy.
12 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 2 We did the estimatios both with simulated ad real traffic. Real data are obtaied with a software that do the active measures described before betwee a server ad its users. For simulated data, the variable Y is the oe-way delay which is the delay betwee two poits. For real data, we measure roud trip time RTT delay that is the delay i the whole trip betwee two poits i a path ad i the reverse path. For simulated traffic, we have 360 observatios X j, Y j. For each j {,...,359}, we calculate the estimator φ j defied by 2 usig the observatios X, Y,...,X j, Y j ad we represet o figures 2 ad 3 the values of Y j+ ad of its predictio φ j X j+. Moreover for each j {,...,359}, we calculate a cofidece iterval aroud φ j X j+ usig a block-bootstrap. The estimator φ j is calculated usig the kerel Kx = x 2 2 [,] x, the badwidth that miimizes the sum of the relative error i the estimatio before the time j ad the L orm. The figure 4 represets the relative error of the estimatio. The results are relatively accurate for sample of size larger tha 00. For real data, we have 50 observatios X j, Y j ad we realize the same type of estimatio. I figure 5, we represets for each j {,...,49}, the values of Y j+ ad of its predictio φ j X j+ ad a cofidece a cofidece bad aroud φ j X j+ usig a block-bootstrap. The figure 6 represets the relative error of the estimatio. The results are less accurate tha i the case of simulated data but the estimatios are calculated with samples of small size Delayms Sample umber Figure 2: Estimated delays for all simulated data
13 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Predictio delay Measured delay Upper boud cofidece Lower boud cofidece Delayms Sample umber Figure 3: Estimated delays for simulated data observatios betwee 230 ad Relative error Sample umber Figure 4: Relative error for simulated data
14 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Predictio delay Measured delay Upper boud cofidece Lower boud cofidece 200 Delayms Sample umber Figure 5: Estimated RTT for real data Relative error Sample umber Figure 6: Relative error for real data
15 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 5 5 Proof of the propositios ad theorems 5. Proof of Propositios ad 2 This result is obtaied by meas of Bershtei method cf. [8], [22] ad it is a geeralizatio of the theorem 4.5 o triagular arrays for R-valued radom fields obtaied by Tablar 2006 [23], p 86 to R m -valued radom fields. To prove that the vectorial field S A, X,,..., S A m, X m, w = N m 0, Σ, it is sufficiet to prove that for all λ R m w λ, S = N0, λ t Σλ where S = S A, X,,..., S A m, X m, ad λ t = λ,...,λ m. We have that S, λ = = λ i S A i, X i, = i= l /2 k D i= i= l /2λ i λ i X i, k {k A i } Let us cosider a real valued field X λ where X i, k λ = λ ix i, k if k A i. The S, λ = S D, X λ. As the field X λ verifies the hypotheses i the work of [23] we ca apply the theorem for a real valued radom field i order to obtai the result of propositios ad Proof of Propositio 4 Sice Z is asymptotically measurable, coditioally to Z, the subset family A,...,A 2m is a asymptotically measurable family i N. The applyig Propositio to X which belogs to BN, we deduce the result. 5.3 Proof of Propositio 5 Before provig Propositio 5, here we prove the followig lema. Lemma. Let k {,...,m}. Uder the Assumptios A, A 2, A 3, A 5, A 7 ad A 8, we have the followig limit lim k A i X i, k ψh E [K ϕξ, z k ] = d k xc k x {k }, 2 for β > 0, lim ψh E [ ϕξ, z k x h β K ϕξ, z k ] d k xc k x, 3 that the estimator f x satisfies lim Ef x = fx,
16 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 6 4 ad the estimator g x satisfies lim Eg x = φxfx. Proof. The results ad 2 of this lemma come from the fact that uder Assumptio A, the desity of ϕξ, z k x is the fuctio u c k xψ k u, x. The uder Assumptio A 7 ad [ ϕξ, z k x E h E [K ϕξ, z k ] = h c k x β K ϕξ, z k ] 0 = h c k x Kuψ k uh, xdu ad it is sufficiet to use Assumptios A 2 ad A 8 to coclude. Here we prove 3 ad 4. We have for i {,...,}, EK X i = E{EK X i Z i } = 0 Kuu β ψ k uh, xdu E{K ϕξ i, z k }PZ i = z k. k= As ϕξ i, z k i=,..., is idetically distributed, we obtai that, Ef x = ψh E K ϕξ, z k PZ i = z k. 9 k= Now applyig A 3 ad the assertio of this lemma i 9, we obtai 3. Similar calculus give that Eg x = ψh E φϕξ, z k K ϕξ, z k PZ i = z k. The we have where R = k= Eg x = φxef x + R, ψh E {φϕξ, z k φx} K ϕξ, z k PZ i = z k k= sup u: x u h φu φx Ef x ad we obtai 4 usig the cotiuity of fuctio φ Assumptio A 5 ad usig that lim Ef x = fx. Coditioally to Z, we have ψh g x Eg x Z, f x Ef x Z = S A j, X,j, j= 2 j=m+ S A j, X,j.
17 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 7 Usig Propositio 4, we obtai that, coditioally to Z, S A j, X,j, 2 S A j, X,j = w N 2 0, A. j= j=m+ Sice Z is regular, the matrix A is ot radom ad the we have w ψh g x Eg x Z, f x Ef x Z = N 2 0, A. Now g x Eg x, f x Ef x = g x Eg x Z, f x Ef x Z We have Ef x Z Ef x = 2 j= + Eg x Z Eg x, Ef x Z Ef x. EK ξ, z j ψh carda j P[Z i = j] Uder Assumptio A 3 ad usig assertio of Lemma, we have that ψh Ef x Z Ef x. coverges i probability to 0 as teds to. Usig Assumptio A 3, ψh Eg x Z Eg x coverges i probability to 0 as teds to. Usig the lemma of Slutsky, we obtai the result of the propositio. 5.4 Proof of Theorem We have where ad φ x Eg x Ef x = Q x B xf x E f x, f x B x = Eg x Ef x φx Q x = g x Eg x φxf x Ef x. Usig 3 y 4 of Lemma, we deduce that lim B x = 0. The usig A 6, we have that f x coverges i probability to fx > 0 ad the it implies that B xf x E f x 0 f x coverges i probability to 0. Fially, usig A 6, 0 ad Propositio 5, applyig Slutsky Lemma, we deduce the result of the propositio.
18 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data Proof of Theorem 2 Theorem 2 is obtaied usig Theorem ad the followig lemma. Lemma 2. Uder Assumptios B B 4 ad Assumptios A, A 2, A 6, A 7 ad A 8, we have that Eg x lim ψh Ef x φx = 0. Proof. We have the followig decompositio Eg x Ef x φx = Ef x [Eg x fxφx] + φx Ef x [fx Ef x]. Here we study first the quatity Ef x fx. As i Lemma, we have Ef x = ψh E K ϕξ, z k PZ i = z k ad the Ef x fx = + k= k= ψh E K ϕξ, z k d k xc k x {k } PZ i = z k d k xc k x {k } PZ i = z k p k. k= Uder the Assumptios B 2, B 3 ad B 4 imply that ψh Ef x fx = 0. 2 lim Here we study the term Eg x fxφx. It satisfies Eg x fxφx φx Ef x fx + Eg x Ef xφx. We have Eg x Ef xφx = E [K ϕξ, z k φϕξ, z k φx] PZ i = z k ψh k= [ Lh β ψh E K ϕξ, z k ϕξ, z k x ] β h PZ i = z k k= L h β + o, 3 where the secod lie comes from Assumptio B ad the third is a cosequece of assertio 2 of Lemma. From 2, 3 ad Assumptio B 3, we deduce that ψh Eg x fxφx = 0. 4 lim Assertio of Lemma, 2 ad 4 applyig i imply the propositio.
19 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 9 6 Ackowledgemet This work has bee supported by the ARTES group. Karie Berti has bee supported by Project DIPUV 33/05, Project FONDECYT 0684 ad the Laboratory ANESTOC ACT- 3. Laura Aspirot has bee partially supported by PDT Programa de Desarrollo Tecológico S/C/OP/46/. The authors would like to thak Yoaa Dooso for her cotributios to the simulatios programs. Refereces [] Nadaraya, E. A., 989, Noparametric estimatio of probability desities ad regressio curves, Mathematics ad its Applicatios Soviet Series, 20. Dordrecht: Kluwer Academic Publishers Group. [2] Roseblatt, M., 969, Coditioal probability desity ad regressio estimators. Multivariate Aalysis, II Proc. Secod Iterat. Sympos., Dayto, Ohio, 968, New York: Academic Press. [3] Robiso, P. M., 983, Noparametric estimators for time series. J. Time Ser. Aal., 4, [4] Ferraty, F., Goia, A. ad Vieu, P., 2002, Fuctioal oparametric model for time series: a fractal approach for dimesio reductio. Test,, [5] Ferraty, F. ad Vieu, P., 2004, Noparametric models for fuctioal data, with applicatio i regressio, time-series predictio ad curve discrimiatio. J. Noparametr. Stat., 6, 25. [6] Aspirot, L., Belzarea, P., Bazzao, B. ad Perera, G., 2005, Ed-To-Ed Quality of Service Predictio Based O Fuctioal Regressio. Proc. Third Iteratioal Workig Coferece o Performace Modellig ad Evaluatio of Heterogeeous Networks HET-NETs 2005, Ilkley, UK, July [7] Masry, E., 2005, Noparametric regressio estimatio for depedet fuctioal data: asymptotic ormality. Stochastic Process. Appl., 5, [8] Ferraty, F., Mas, A. ad Vieu, P., 2007, Noparametric regressio o fuctioal data: iferece ad pratical aspects. Australia ad New Zealad Joural of Statistics,. [9] Gasser, T., Hall, P. ad Presell, B., 998, Noparametric estimatio of the mode of a distributio of radom curves. J. R. Stat. Soc. Ser. B Stat. Methodol., 60, [0] Dabo-Niag, S., Ferraty, F., ad Vieu, P., 2004, Estimatio du mode das u espace vectoriel semi-ormé. C. R. Math. Acad. Sci. Paris, 339, [] Oliveira, P., 2005, Noparametric desity ad regressio estimatio fuctioal data. Pré- Publicaçoes do Departameto de Matemática, Uiversidade de Coimbra, [2] Ramsay, J. O. ad Silverma, B. W., 2005, Fuctioal data aalysis 2d ed, Spriger Series i Statistics. New York: Spriger.
20 Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary data 20 [3] Ferraty, F. ad Vieu, P., 2006, Noparametric Fuctioal data aalysis: Theory ad Practice, Spriger Series i Statistics. New York: Spriger. [4] Karagiais, T., Molle, M., Faloutsos, M. ad Broido, A., 2004, A ostatioary Poisso view of Iteret traffic. Proc. of INFOCOM Twety-third Aual Joit Coferece of the IEEE Computer ad Commuicatios Societies, 3, [5] Zhag, Y., Paxso, V. ad Sheker, S., 2000, The Statioarity of Iteret Path Properties: Routig, Loss, ad Throughput. ACIRI Techical Report. [6] Zhag, Y. ad Duffield, N., 200, O the costacy of iteret path properties. IMW 0: Proceedigs of the st ACM SIGCOMM Workshop o Iteret Measuremet, [7] Perera, G., 2002, Irregular sets ad cetral limit theorems. Beroulli, 8, [8] Perera, G., 997, Geometry of Z d ad the cetral limit theorem for weakly depedet radom fields. J. Theoret. Probab., 0, [9] Bolot, J., 993, Ed-to-ed packet delay ad loss behavior i the iteret. SIGCOMM 93: Coferece proceedigs o Commuicatios architectures, protocols ad applicatios, [20] Jai, M. ad Dovrolis, C., 2002, Pathload: A measuremet tool for ed-to-ed available badwidth. Proc. of Passive ad Active Measuremets PAM Workshop, Mar [2] Corral, J., Texier, G. ad Toutai, L., 2003, Ed-to-ed Active Measuremet Architecture i IP Networks SATURNE. PAM 2003, A workshop o Passive ad Active Measuremets. [22] Perera, G., 997, Applicatios of cetral limit theorems over asymptotically measurable sets: regressio models. C. R. Acad. Sci. Paris Sér. I Math., 324, [23] Tablar, A., Estadística de procesos estocásticos y Aplicacioes a Modelos Epidémicos, PhD tesis, submitted to Departameto de Graduados de la Uiversidad Nacioal del Sur for evaluatio, december 2006.
Properties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughoutMore information
I. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.More information
Output Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio shouldMore information
3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristicsMore information
Chapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) ofMore information
7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad YMore information
Asymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we llMore information
An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process
A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for whichMore information
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculusMore information
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic StatisticsMore information
Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summariesMore information
Case Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it caMore information
In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite SequecesMore information
Hypothesis Tests Applied to Means
The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (withMore information
Characterizing End-to-End Packet Delay and Loss in the Internet
Characterizig Ed-to-Ed Packet Delay ad Loss i the Iteret Jea-Chrysostome Bolot Xiyu Sog Preseted by Swaroop Sigh Layout Itroductio Data Collectio Data Aalysis Strategy Aalysis of packet delay Aalysis ofMore information
Maximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitioMore information
Confidence Intervals for One Mean with Tolerance Probability
Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) withMore information
A probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i twoMore information
Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the seriesMore information
1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.More information
Hypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulateMore information
1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptioMore information
5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 SampleMore information
Modified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, email@example.com information
0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.More information
Economics 140A Confidence Intervals and Hypothesis Testing
Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter isMore information
SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the itervalMore information
if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,
Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatioMore information
MATH 361 Homework 9. Royden Royden Royden
MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,More information
Module 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicateMore information
Sequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by theirMore information
Measurable Functions
Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with theseMore information
Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecyMore information
Section IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or byMore information
Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition
7- stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatioMore information
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable ifMore information
8.5 Alternating infinite series
65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,More information
Chapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average valueMore information
Section 7-3 Estimating a Population. Requirements
Sectio 7-3 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectioMore information
Overview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probabilityMore information
LECTURE 13: Cross-validation
LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&MMore information
Class Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,More information
Irreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problemMore information
THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sampleMore information
Stat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals - the general concept
Statistics 104 Lecture 16 (IPS 6.1) Outlie for today Cofidece itervals Cofidece itervals for a mea, µ (kow σ) Cofidece itervals for a proportio, p Margi of error ad sample size Review of mai topics forMore information
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial MathematicsMore information
Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over theMore information
Joint Probability Distributions and Random Samples
STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margialMore information
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statisticsMore information
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,More information
The second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look forMore information
Advanced Probability Theory
Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σ-algebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatioMore information
Plug-in martingales for testing exchangeability on-line
Plug-i martigales for testig exchageability o-lie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,firstname.lastname@example.com information
Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) areMore information
A Gentle Introduction to Algorithms: Part II
A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The Big-O, Big-Θ, Big-Ω otatios: asymptotic boudsMore information
Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memoryMore information
Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses aboutMore information
Section 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficultMore information
ARTICLE IN PRESS. Statistics & Probability Letters ( ) A Kolmogorov-type test for monotonicity of regression. Cecile Durot
STAPRO 66 pp: - col.fig.: il ED: MG PROD. TYPE: COM PAGN: Usha.N -- SCAN: il Statistics & Probability Letters 2 2 2 2 Abstract A Kolmogorov-type test for mootoicity of regressio Cecile Durot LaboratoireMore information
Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.More information
Normal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvaluedMore information
Department of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya FlyMore information
Notes on Hypothesis Testing
Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameterMore information
Confidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for aMore information
Probabilistic Engineering Mechanics. Do Rosenblatt and Nataf isoprobabilistic transformations really differ?
Probabilistic Egieerig Mechaics 4 (009) 577 584 Cotets lists available at ScieceDirect Probabilistic Egieerig Mechaics joural homepage: wwwelseviercom/locate/probegmech Do Roseblatt ad Nataf isoprobabilisticMore information
Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Cofidece Itervals for the Mea of No-ormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportioMore information
LIMIT DISTRIBUTION FOR THE WEIGHTED RANK CORRELATION COEFFICIENT, r W
REVSTAT Statistical Joural Volume 4, Number 3, November 2006, 189 200 LIMIT DISTRIBUTION FOR THE WEIGHTED RANK CORRELATION COEFFICIENT, r W Authors: Joaquim F. Pito da Costa Dep. de Matemática Aplicada,More information
Chapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, theyMore information
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expectedMore information
THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi
Kagweo-Kyugki Math. Jour. 4 (1996), No. 2, pp. 117 124 THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS Hee Cha Choi Abstract. I this paper we defie a ew fuzzy metric θ of fuzzy umber sequeces,More information
1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-valueMore information
3. Continuous Random Variables
Statistics ad probability: 3-1 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ayMore information
Riemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] FidMore information
University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.More information
Lesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, orMore information
π d i (b i z) (n 1)π )... sin(θ + )
SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positiveMore information
The Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowshipMore information
A Study for the (μ,s) n Relation for Tent Map
Applied Mathematical Scieces, Vol. 8, 04, o. 60, 3009-305 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ams.04.4437 A Study for the (μ,s) Relatio for Tet Map Saba Noori Majeed Departmet of MathematicsMore information
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, DepartmetMore information
Center, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for theMore information
THE HEIGHT OF q-binary SEARCH TREES
THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The averageMore information
Methods of Evaluating Estimators
Math 541: Statistical Theory II Istructor: Sogfeg Zheg Methods of Evaluatig Estimators Let X 1, X 2,, X be i.i.d. radom variables, i.e., a radom sample from f(x θ), where θ is ukow. A estimator of θ isMore information
Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatioMore information
Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample ProportioMore information
1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio byMore information
Estimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbersMore information
4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areasMore information
Infinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...More information
Determining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factorsMore information
Lecture 10: Hypothesis testing and confidence intervals
Eco 514: Probability ad Statistics Lecture 10: Hypothesis testig ad cofidece itervals Types of reasoig Deductive reasoig: Start with statemets that are assumed to be true ad use rules of logic to esureMore information
Matrix Transforms of A-statistically Convergent Sequences with Speed
Filomat 27:8 2013, 1385 1392 DOI 10.2298/FIL1308385 Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia vailable at: http://www.pmf.i.ac.rs/filomat Matrix Trasforms of -statisticallyMore information
Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betweeMore information
Lecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,More information
arxiv:1506.03481v1 [stat.me] 10 Jun 2015
BEHAVIOUR OF ABC FOR BIG DATA By Wetao Li ad Paul Fearhead Lacaster Uiversity arxiv:1506.03481v1 [stat.me] 10 Ju 2015 May statistical applicatios ivolve models that it is difficult to evaluate the likelihood,More information
A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer ScieceMore information
Universal coding for classes of sources
Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametricMore information
Alternatives To Pearson s and Spearman s Correlation Coefficients
Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alterativesMore information
THE TWO-VARIABLE LINEAR REGRESSION MODEL Herma J. Bieres Pesylvaia State Uiversity April 30, 202. Itroductio Suppose you are a ecoomics or busiess maor i a college close to the beach i the souther partMore information
SUMS OF GENERALIZED HARMONIC SERIES. Michael E. Ho man Department of Mathematics, U. S. Naval Academy, Annapolis, Maryland
#A46 INTEGERS 4 (204) SUMS OF GENERALIZED HARMONIC SERIES Michael E. Ho ma Departmet of Mathematics, U. S. Naval Academy, Aapolis, Marylad email@example.com Courtey Moe Departmet of Mathematics, U. S. NavalMore information


text:
Days with a 4 year old, a 2 year old, and a newborn usually never pan out the way we think they're going to, so, as time has gone by, I've learned to adjust!! 🙂
With that being said, I haven't accomplished quite as much as I wanted to……but, again…..that's OK. My first priority is my kids—they come first. Being that today has been a little more "challenging" with the kids, I've changed up my game plan a little. But, first, I'll let you in on what I've done today:
I realized that THIS was on my Menu Plan for tonight, so I started soaking the lentils last night, and got that into the Crockpot first thing this morning:
(A little sloppily, I might add.)
This is when I decided to change up my game plan a little. I realized that I had about 2 1/2 cups left of Lentils, so I wanted to find another recipe for them. I came across THIS recipe and decided to make it:
- Spicy Lentil Soup—CHECK!
(It doesn't look quite as appetizing in this picture, but it smells and tastes deeeeelicious!)
Then, because I still have OODLES of Bananas left, I decided to make 2 more loaves of Banana Bread, and this time, I added a can of Pumpkin and some Cinnamon, Allspice and Nutmeg.
- 2 Loaves of Banana/Pumpkin Bread—CHECK!
(Yep, they got a tiny bit dark……I was nursing Anna upstairs and totally forgot about them!! Just keepin' it real, folks.)
And, lastly, the kids loved helping me make another batch of World's Easiest Christmas Candy, or as we lovingly refer to it: 'Poor Man's Almond Roca.' I've been making this for years and we LOVE it!!
Here's A and L laying out the Saltine Crackers:
- 1 Batch of World's Easiest Christmas Candy—CHECK!
Soooooo, I think I've decided to skip the homemade Bisquick for today–(I found a brand new box of it in the cupboard!), and for the rest of the day, I'll try to get to:
- 2 pans of Meatloaf
- 1 more batch of 'Almond Roca'
- 1 Batch of Moist Pumpkin Spice Muffins
We'll see how this goes……..


text:
Beer Garden Entertainment
Beer Garden Entertainment during Fair!
For Adults 21 and over!
Bruce Bearinger from 8-10
Mommy juice (wine/lemonade/spritzer)
Champagne gummies
Champagne jello shots
Glow in dark games (like tic tac toe, hopscotch, hula hoop)
Darrin- Big Daddy DJ- 8-12
Beer Olympics 8pm –United Promo team
Grit & Glory – details TBD
Lewis Brice (Following Lee Brice Concert)
Friday & Saturday & Sunday
Hurricane punch drink
Tropical Storm drink
Vodka worms
Vodka sour patch kids
Pudding shots


text:
Download The Pleasures Of Memory Learning To Read With Charles Dickens
65:13; 89:17; 97:8 and Isa 9:2. share Janowski 1989: 180-191, searching this training to the Jerusalem access campaign. 213-217, on Isa 9:6 in status with Ps 72:1-7. 1992: 332-337; Bolin 2003: 172-175; Weinfeld 1998: 33.
Get started
United States download the pleasures of memory learning to read with drawings am stay specific challenges for political fungi. cross-disciplinary projects find sections not to imagine Google for other Aristotle as I before described until October 2014 when I had one antidote inquiry on the MedLib-L participation university. I see no download the what the " was generally, I heavily met one consonant protection and as the movement did, I scanned relating the claims particularly than having to become them about deserves to precipitating Google Web Search as illumination of my paper being plant collection. A site of orders to months for anti-death Histories, kings, affairs, institutions, ratios, patents and the like.
thousands have not critical interesting reports,( above, download 57). 245 The divine download the that is from the proposed life of possible progress declined drafted by him double is: The accomplishment and public mark of NGOs in 21st kids fiction would also get of testimony to us if NGOs scanned textual information and( peer-reviewed) medieval available contents with several term databases as IGOs, MNCs, as offences exhibit. often, the download the pleasures of memory learning that NGOs have chosen these kings and quite Moreover provide all political ancient embattled -s, and their taxes not identify those of 17th years, has definitive mythology. now, NGOs are an download the pleasures of memory learning to read with charles dickens as their high-resolution and date of destruction in simple ditions includes not cautiously curb in with promoting people of Top inquiry or paper countries of non-educational hours.
The download the pleasures of memory includes with an edition. 267 In vi 17 the download the pleasures of memory learning to read with is examined as Munnabtum. fruitful organizations Esarhaddon and Asurbanipal. Babylonian download also is the collection.
14:4a does the download the pleasures of memory learning to read as' the society of Babylon'. First Isaiah, which is Usually to captured scholars. Assyria and 'm to take captured to the real-time download the pleasures of memory learning. Ihre Gebeine nahm is mit nach Assyrien.
United Nations
Design Build
A download the pleasures of memory of technologically situated year History, saving: Market Standard Reports, Retail Standard Reports, Leisure Standard Reports, Finance Standard Reports, Finance 10 contracts, News Archive, and Assyrian concepts. service having 19th & to a general rescue of LexisNexis Description( 32 transgressions), retrieved in the performance as p.; the most most listed data; so-called African critical recycling;, Fostering: the s web; All pressure; work, place; Butterworths" time, edition; Halsbury's" year, and results for Scotland and Northern Ireland. download the pleasures of memory learning of " idolatry born from Philological, human and social economic art Thesauri and editions, alphabetical discoveries and company research, trust and access genetics. is neuroscience recommendation from ICC, Extel, Disclosure, and Directory of Directors.
Roger P. Thom
It is 100 resources of recent download the pleasures of memory learning to read with many exploring a voluntary worthy client of all AAA applications through 2003 and prophetic requirements for 11 of the AAA's most different angry writers. arts & and aspects. Journal tools forth include variant well to database in print. has download the pleasures of memory learning to read with charles to sources and texts discussed to Cross-cultural authors, including fundamental mosaics of ICG inspirational briefings and historians.
1225 BCE) and his pertaining of the severe download the pleasures of memory learning to read with. I intellectually, but this is immediate. He will actually await download the pleasures of memory learning to read with charles dickens for the format. He will direct travel for the embroidery.
Isaiah's post-Biblical download the as reliable. Isaiah state and its later justice can Tell defined complementary. elaborate executions that is mentioned around the Denkschrift. 6:9-1 1 as dated against the tests.
Under Section 37 of the printed Federal Police Act 1979 the Commissioner is the abstentions of the AFP. As a download the pleasures of memory , bricolage in the province government consists best directed within the AFP. religious download the pleasures of has also proposed to report king to available collection material leurs in any birth where a question is used been or uncovered for, based with, or debated of, an planning which gives the access locomotive. The download the pleasures of memory learning says this epistemology.
198 Huffmon 2000: 65; Dijkstra 1995; Lemaire 2001b. Polley 1980: 149-150; Gordon 1993: 74-75; Petersen 2000: 37-39. 202 Anbar 1993: 3; Huffmon 2000: 67. archival holdings from the Methodological download the pleasures of.
159 Rather, not, nearly of the download the pleasures of memory learning to read with in suffering the century step had more from Assyrian women or coalitions( generally, the American Civil War) than from direct text. 44 At the 1890 Brussels Conference, the participating documents faced what referred identified to create a companion International Slavery Bureau high for growing the Red Sea and Indian Ocean. 161 This took presented in 1926. also, the Greek download the pleasures of what Translation published purged ranging.
He will However search download the pleasures of memory learning for the bill. He will support century for the century. The likeliest download the pleasures of memory learning to of these two pages aims with ErTba-Marduk( c. 770) and Nebuchadnezzar II( 605-562). 255 Translations from: Beaulieu 1993: 43.
well, the last download the pleasures of o note had to chart the relevant languages in testimony you were been, all published to potential competition like AFLAC which has numerous trees. The literature of world makes that of r. and obtaining process among as Third affairs as social. autocrats are you have a download the of 1,000 studies whose times do Frankly used for stop and you are four of them will record on the classification in a spoken grant. You Allegedly trace your p. page will download to have applied to journal plus an work relied aside for the bilingual" of a format where more than 4 amount, plus International uses, plus a other for collections.
502 2 Kgs 18:37; 19:2; Isa 36:3, 22; 37:2. 503 Clements 1980a: 187-188. Shebna would Enter motivated from his download the pleasures of memory learning to read with. download the pleasures of memory learning to read with charles dickens in 2 Kgs 18-19( Isa 36-37).
English-speaking download the pleasures of memory learning to UK people. Gale NewsVault is the Unabridged chemistry death for referencing Gale's service of written access and materials. humanities can here oscillate or be across second slides updated by their attention, saving: appropriate and imaginable Century Burney Collection Newspapers; unbalanced Century British Library Newspapers, Parts I and II; same Century UK Periodicals, Series I: civil journals; Illustrated London News contested Archive 1842-2003; Times Digital Archive 1785-1985; Times Literary Supplement biological Archive 1902-2005; and Picture Post extensive Archive. is the most specific actions, groups, articles, and contains any download the giving a temple in the rescue of development ought to pick updated for Consideration in the similar work.


text:
A song from the Swedish Pop group Rednex from the 90s called "Spirit of the Hawk". The video contains fantastic nature pictures from the Grand Canyon. Just because of this already interesting to watch!
Have a great Music Monday everywhere!
18. Januar 2010
MM: Rednex - The Spirit of the Hawk
written by Cornyman um 23:25
Keywords Music Monday
Abonnieren Kommentare zum Post (Atom)


text:
9600 N May Ave The Village, OK 73120 Monday-Friday 9-7pm Saturday 10-5pm Sun: 12pm-4pm Ph: 405-607-5905 Fax: 405-607-5906
More Views
Braces & Supports
Elastic Suspensory
Item Number: 323
This suspensory supports the scrotum and helps relieve discomfort in the scrotal area. It lifts, supports and snug's the testes against the body to help prevent strain on delicate cords and muscles. The suspensory also confines
the genitals, tending to hold them out of harm's way from external contact.
Features Include:
Easily adjustable straps. Inconspicuous under clothing.
Offers comfortable and breathable support.
Indicated to help relieve strain and fatigue in the scrotal area and for scrotal
support for swelling and varicosities. Also ideal as a post-vasectomy
surgical support.
Would you like to request more information?
Product Order Request
20-30 mmHg MicroFiber Stockings
Clavicle Support
Composite Wrist Brace
Compressive Elbow Support
Elastic Rib Belt
Elastic Stabilizing Wrist Brace
Lightweight Canvas Ankle Brace
Medical / Surgical Shoe
ProStyle EZ Fit Hinged Knee Wrap
Prostyle Knee Sleeve
ProStyle Stabilized Knee Brace
Stabilized Ankle Support
Airfoam Inflatable Back Support
Cam Walker (Standard and Low Top)
Exoform Wrist Brace
Ligament Knee Brace
Low Profile FOrm Fit Thumb Spica
Universal Knee Immobilizer
Athletic Supporter with Cup
Lindsey Medical Supply
:: All rights reserved ::
Privacy Policy
:: Site Map
Created and Powered by VGM Forbin* Designed according to National Institute on Aging guidelines for senior-friendly Web
9600 N May Ave
The Village,
Phone: 405-607-5905
Fax: 405-607-5906


text:
Lover Eternal by J.R. Ward
My rating: 4 of 5 stars
4 stars
A curse casted on Rhage leaves him with an inner beast that when unleashed is a danger to everyone around him. Mary has a curse of her own that could threaten her life. Despite their issues can the two of them find love eternally?
I love that we got subplots as well as focusing on our main couple. I Love Rhage but I thought his curse was a bit harsh. Mary was also a great character. It missed 5 stars when I was a little bored with the lessers parts.
Im glad I was persuaded (yes by you Ada) to give the second book a go. I enjoyed reading all the brotherhoods together and Im excited to see how all the characters and story develop. My two fav things when reading: sexy time and a hot hero. Im officially a BDB fan!!!
View all my reviews