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Original question:
In the Wikipedia article on the Rao–Blackwell theorem, we read:
In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased estimator of zero", the Rao–Blackwell process is idempotent.
and indeed this is commonly heard and quite easy to prove (just apply definitions).
Suppose the hypothesis of completeness were dropped. What would be an example in which idempotence would not hold?
Later edit for the sake of self-containment:
Consider a parametrized family of probability distributions of a random variable $X$, with parameter $\theta$. In typically seen examples $X$ is a sequence $X_1,\ldots,X_n$ of independent identically distributed random variables and $\theta$ is a tuple of one or two scalars. $\theta$ may be $(\mu,\sigma)$, the expectation and variance of a normal distribution, or may be $p$, the expected value of a Bernoulli-distributed random variable (thus e.g. the proportion of the population who have blue eyes), or may be the shape and scale parameters of a gamma distribution, etc.
A statistic is an "observable" random variable, i.e. its value depends on $(X,\theta)$ only through $X$. E.g. heights of 21-year-old men may be normally distributed with expectation $\mu$ and standard deviation $\sigma$; we observe the heights of $20$ such men, randomly chosen. The average of the $20$ heights is observable; the amount by which that average exceeds the population average $\mu$ is unobservable. Often unobservable things are those that depend on the values of parameters represented by Greek letters.
A sufficient statistic is a statistic $T(X)$ such that the conditional distribution of $X$ given $T(X)$ does not depend on $\theta$.
For example:
- If $X_1,\ldots,X_n\sim \operatorname{i.i.d.} N(\mu,\sigma^2)$, it can be shown that the conditional distribution of $X_1,\ldots,X_n$ given the pair $(X_1+\cdots+X_n,X_1^2+\cdots+X_n^2)$ does not depend on $(\mu,\sigma^2)$. Thus that pair of random variables is a sufficient statistic for that family of probability distributions.
- If $X_1,\ldots,X_n\sim\operatorname{i.i.d.} \operatorname{Uniform}(0,\theta)$, then the conditional distribution of $X_1,\ldots,X_n$ given $\max\{X_1,\ldots,X_n\}$ does not depend on $\theta$. Thus $\max\{X_1,\ldots,X_n\}$ is a sufficient statistic for that family of distributions.
- If $X_1,\ldots,X_n\sim\operatorname{i.i.d.}\operatorname{Gamma}(\alpha,\beta)$ (i.e. with density proportional to $x\mapsto x^{\alpha-1}e^{-x/\beta}$ for $x > 0$), then the conditional distribution of $X_1,\ldots,X_n$ given $(X_1+\cdots+X_n,\ X_1\cdots X_n)$ does not depend on $(\alpha,\beta)$. Thus the pair whose components are the sum and the product is a sufficient statistic for that family of distributions.
- If $X$ has density proportional to $x\mapsto e^{-|x-\theta|}$ for $x\in\mathbb{R}$, and the parameter space is given by $\theta\in\{1, -1\}$, then $$\begin{cases} 1 & \text{if }X\ge 1 \\ -1 & \text{if }X\le -1 \\ X & \text{if }-1<X<1 \end{cases}$$ is a sufficient statistic.
Intuitively, a sufficient statistic constains all information in the data that is relevant to estimating the value of $\theta$, if the model, which is the parametrized family of disetributions, is right. But a sufficient statistic contains none of the information in the data that might indicate that the model is wrong (unless what's wrong is the support of the distribution; e.g. you get a negative number when you thought the Gamma family was right).
A complete statistic is a statistic $S(X)$ (i.e. and observable random variable) that admits no unbiased estimator of $0$. An unbiased estimator of $0$ is a function $f(S(X))$, where $f$ is not allowed to depend on $\theta$ whose expected value remains equal to $0$ regardless of the value of $\theta$. Thus in the first example above, $(X_1,X_2)$ is a statistic, but is not complete, since the expectation of $X_1-X_2$ is $0$ regardless of the values of $\mu$ and $\sigma^2$. The sum $X_1+\cdots+X_n$ is a complete statistic in that case.
An unbiased estimator of $g(\theta)$ is a statistic (an observable random variable) whose expected value remains equal to $g(\theta)$ regardless of the value of $\theta$. (The emphasis on "observable" is a sort of warning against a newbie mistake that says a biased estimator can be changed to an unbiased estimator by subtracting the bias from it. That doesn't work since the bias is unobservable in all cases where one would need to seek an estimator. The resulting random variable is not a statistic; it cannot be used as an estimator.)
The Rao–Blackwell theorem says that if $h(X)$ is an unbiased estimator of $\theta$ and $T(X)$ is a sufficient statistic for $\theta$ (i.e. for the family of distributions parametrized by $\theta$) then $j(X)=\mathbb{E}(h(X) \mid T(X))$ is an unbiased estimator that is no worse, in the sense of mean-square-error or any of a variety of similar function, than is $h(X)$. In typical applications, $j$ is far better than $h$, and that's why Rao–Blackwell is used.
For example: suppose one wants a good unbiased estimator of the probability that no phone call comes into the switchboard in the next two minutes, when their arrival is a Poisson process with a rate of $\lambda\text{ per minute}$. The parameter $\lambda$ is unobservable. One wants to estimate $e^{-2\lambda}$. The data consist of numbers of calls that arrived in each of $10$ disjoint two-minute intervals in the past $20$ minutes. A crude unbiased estimator is $1$ if no call arrived in the first two minutes and $0$ if one or more arrived. Exercise: Find the conditional expected value of that crude estimator given the data. You should get $\left((9/10)^\text{(number of calls in the past 20 minutes)}\right)$. That is the improved unbiased estimator.
You start with a crude estimator $h(X)$ and find a better estimator $j(X)$. What happens if you apply the process again, starting with $j(X)$?
If the sufficient statistic happens to be complete, you just get $j(X)$ again. That's idempotence. The proof of idempotence is just definition-chasing.


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Ideas for Blog Posts: Inspiration When You Have NoneAlli Worthington
It's never easy to blog consistently. Sometimes you just "aren't in the mood" and can't figure out where to draw inspiration from. Let these ideas help you when you just aren't feeling the blogging magic.
You can choose one idea a week, or one idea a day, or use them to spark your own blog post ideas. Jot down your ideas when you get them or DM them to yourself like I do. Or Skype. It doesn't matter how you do it, just do it. Ready? Let's go.
Posts You Can Write Off The Top of Your Head
Screenr Video to Start the Post Maybe you just need to do a short video post instead of trying to write first. Screenr is a free way to show your computer screen. My friend Angela England uses this to show simple how-tos on her blogging blog and likes it because the videos can only be 5 minutes maximum. Once the screencast is finished you can write in the post part based on what you talked about!
Example – How to Align Images in WordPress on AngEngland.com
Top Ten Tips For ________ – Whatever you've been talking about lately. Fill in the blank and get writing! "Procrastinating on a Blog Post" or "Stopping Internet Abuses" are both perfectly valid posts in my mind. Don't stress about this too much. Whatever the first thing that pops into your head is, start writing it! Heart dump because sometimes those are the best pieces.
Example – Top Ten Crafts for Preschoolers and Young Children on Blissfully Domestic
Round Up of Your Best Work — Why not highlight yourself and some of your most remarkable stuff? Do a round-up of the most commented on blog posts. Or your most popular pieces ever. Or maybe it's just the posts that you thought were super fabulous even if no one else really picked up on them. "Posts Y'all Should Have Loved" sounds like an awesome blog post to me!
Example – How to Write Great Blog Content on Problogger
Do you have a favorite source of blogging inspiration? Share it with us in the comments and you may be featured in a future Blog Ideas post!

subset: commoncrawl


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Crash! Clamor! Trumpet blasts jolted the Midianite army from their sleep.
Stunned and confused by the chaos of sound and lights, the Midianites scrambled for their weapons. In the darkness and confusion, the soldiers panicked, thinking they were outnumbered.
God caused each man to believe that his comrade was an Israelite. Midianite clashed sword with Midianite, while others fled for their lives. They used their swords to provide a way of escape, killing their fellow soldiers by mistake.
"We're under attack! The camp is under attack!"
Seeing their camp surrounded by torch lamps on three sides, the Midianites thought tens of thousands of Israelite soldiers were about to swoop down upon them.
The groggy Midianites sprang up, frightened by the sound of the trumpet blasts, which they knew was the signal for war. Those who did not turn their swords on one another fled.
"Israel is about to overtake us—we're all doomed!"
In reality it was only 300 Israelite soldiers, led by Gideon.
Gideon then organized soldiers from the tribes of Naphtali, Asher and all of Manasseh to pursue the retreating Midianite army.
Soon, God had granted victory to the Israelites. And after defeating the retreating armies, the Israelites cried for Gideon to be their king. He told them to instead "serve the Eternal."
But how did Gideon get to this point—having the faith to lead an army of 300 against the Midianite hordes? What qualities allowed God to use him as a leader?
In the village of Ophrah, in the territory of Manasseh, lived Gideon, who was threshing wheat in a winepress. Normally, this was done by using cattle to tread the wheat on a threshing floor. But Gideon did so with a winepress because he feared the Midianites. He knew that they might come at any moment and take away his wheat. Gideon and his family desperately needed this crop, for there was very little wheat left to harvest.
For seven years, God allowed Israel to suffer under the brutal hand of the Midianites and their allies of wandering nations because they did not obey Him. In fear, the Israelites hid in caves.
Gideon worked behind a tree so as not to be seen. As he raced to get the wheat ready so he could hide it from the Midianites, he did not notice a stranger had sat beneath the tree. The stranger appeared to be a handsome, strong-looking man who wore fine clothing, clean and crisp. He definitely did not look like he had been living in a dirty cave.
The stranger said to him, "The Eternal is with you, Gideon, you mighty man of valor!"
Struck by the high quality and boldness of the stranger's appearance, Gideon wondered if he was in the presence of someone important.
Gideon replied, "My lord, if the Eternal is with us, then why are we cowering in caves like wild animals? Why do we flee at the first sight of Midianite troops? Where are all of God's miracles, which our fathers told us about—how God brought us out of Egypt and freed us from the bonds of slavery? The Eternal has forsaken us; He has delivered my people into the hands of the Midianites."
Gideon had grown tired of living in daily fear, feeling helpless and without hope.
The stranger boldly said, "The Eternal is with you. Go in His might and you shall deliver Israel from the Midianites! Have I not sent you?"
Gideon suddenly realized that he was speaking with the Messenger of the Eternal, who would later be known as Jesus Christ.
"But how can I deliver Israel?" Gideon said. "My family is the least in the tribe of Manasseh, and I am the least in my father's household. There's nothing special about me."
Unlike most people, Gideon did not think that he was better than others. This meant that he had the right kind of attitude, the kind that God looks for when choosing His leaders. God rejects those who think they are greater than others—He knows that such vain people will not fear and obey Him.
Out of the tribes of Israel, Gideon's clan, the Abiezrites, was the smallest in Manasseh, which is why Gideon did not think he was great or special.
The Messenger said, "I will be with you, Gideon. I shall enable you to conquer the Midianites as though they were only one man" (Jdg. 6:11-19).
Like Moses before him, Gideon needed to be reassured that God would truly work through him. Gideon could not yet bring himself to believe that God would choose to use someone like him to deliver Israel. So Gideon asked for a sign.
"Please stay here where you are. I will be right back and bring You an offering."
The Messenger graciously agreed, knowing that men need His help to walk by faith and not by sight (II Cor. 5:7).
Gideon quickly gathered the best that he had, a young goat, unleavened bread and broth in a pot—placing it on a rock as directed by the Messenger.
When the Messenger of the Eternal took His staff and touched the offering, fire shot out from the rock, totally consuming the offering!
Amazed by this miracle, Gideon stood in awe. It showed that God had accepted His offering. Gideon turned to speak with the Messenger of the Eternal, but He had disappeared.
Suddenly realizing that this truly was God's Messenger, Gideon feared that he did not pay God's Messenger proper respect.
As he worried, the voice of God spoke to him: "Peace be with you, Gideon. Do not fear. You shall not die."
Thankful and filled with awe, godly fear and respect, Gideon built an altar to God. He called it "The-Eternal-Is-Peace." In the wonderful world tomorrow, all people will learn that living God's way always brings peace (Jdg. 6:20-24).
That same night, God spoke to Gideon again. He said, "Take ten of your trusted servants and take your father's seven-year-old bull with you. Destroy the altar of Baal, which belongs to your father, and chop down the carved image of wood next to it. Cut up the wood and build an altar for Me on the summit of this stronghold. Then use the wood to burn the bull upon My altar."
Knowing the task would be dangerous, Gideon decided to carry out God's instructions under the cover of nightfall. He took ten of his family's most trusted servants and did as the Eternal had commanded.
The next morning, the people of the village were horrified to see what had been done to Baal's altar. They also noticed that the wooden image of the Canaanite goddess Asherah, which stood next to the altar, was missing. The townspeople were shocked and angered.
"Who did this evil deed? Who would be wicked enough to disrespect the holy things of Baal and Asherah?"
Soon, the blame fell on Gideon—and the crowd wanted his life.
Joash, Gideon's father, stood up for his son: "Will you fight Baal's battles for him? If Baal is a god, then let him fight for himself."
The accusers paused, exchanging nervous and embarrassed looks with one another. If they insisted on executing Gideon, it would appear that their pagan god was too weak to deal with a physical man. So they backed away from their accusations and let Gideon be.
News of Gideon's "crime" reached the ears of the Midianites, Amalekites and the other desert wanderers. Fearing they had a rebellion on their hands, the leaders of these pagan hordes set up camp in the Jezreel valley.
Hearing about the approaching threat, Gideon asked God to show him what to do.
God sent the power of His Spirit to give Gideon the courage and boldness to meet the enemy head-on. Wasting no time, Gideon blew the trumpet alarm, calling for his clan, the Abiezrites, to assemble.
Once they had gathered before him, Gideon addressed them with a rousing speech: "For seven years, we have cowered under the threat of the Midianites and their ruthless allies. They have preyed upon our people like locusts. The time has come for our oppression to end."
"But how?" someone shouted.
"By turning back to God—by fearing and obeying Him, and putting the Eternal first in our lives, as our fathers did in the past. God is the reason why Israel had so much success over their enemies, years ago. God will fight for us again, IF we put Him first. But if you continue to bow down to Baal, Israel is doomed. Baal cannot save us. Neither can the other false gods of the land. But the Eternal can!"
With one voice, the Abiezrites agreed to follow Gideon into battle.
Then Gideon sent messengers throughout Manasseh and the neighboring tribes of Asher, Zebulun and Naphtali. And these tribes responded to Gideon's call to fight (Jdg. 6:33-35).
As Gideon and his leaders examined terrain maps and discussed battle plans, men whom he had sent out to spy on the Midianites' camp came back with their report: "The enemy has amassed an army so large, it seems like a swarm of locusts. The Midianites and their allies are spread out as far as the eye can see" (Jdg. 7:12).
The report shook Gideon's faith; he began to have doubts. "Is God truly using me?" he wondered to himself. "I have never commanded an army before. And I come from the smallest clan in my tribe—surely God wants someone greater than me to deliver Israel. After all, who am I?"
At this point, Gideon did not understand that God can use anyone to lead His people—as long as that person is humble and yielded to His divine will.
When he was alone, Gideon prayed to the Eternal. "Almighty God, if You will deliver Israel through me, as You have said, then please prove this to me with a miraculous sign. I shall leave a fleece of wool on the threshing floor overnight. Tomorrow morning, if there is dew on the fleece but the ground is completely dry, then I will know that you shall use me to deliver Israel."
God has often had to encourage His servants, who sometimes lack the faith to go forward. So the Eternal was very patient and merciful to Gideon.
The next morning, Gideon discovered that the fleece was soaked with dew—in fact, he was able to wring out an entire bowl of water from it. Yet the ground had stayed dry. God had answered Gideon's request.
Though encouraged by this miraculous sign, Gideon's faith still was not strong enough. So he prayed to God again: "O Eternal God, please do not be angry with me. Just let me ask of You one more sign: I will leave the fleece on the threshing floor overnight again. When tomorrow morning comes, let the fleece be dry but the ground be wet with dew. Then I will know that You are truly using me."
Again, God did as Gideon requested. The next morning, Gideon anxiously awoke to find that the ground was soaked with dew while the fleece was completely dry!
Gideon was so encouraged by this second miracle that he found the faith to do what had to be done (Jdg. 6:36-40).
Since God had patiently allowed Gideon to test him, God decided that it was time to test Gideon. God often tests His servants to discover their weaknesses and to build His holy, righteous character in them.
Though the Israelites were already outnumbered—32,000 Israelites against "swarms" of enemies, God decided to shrink the size of the army. This was done to test Gideon's faith, as well as to prove, without a doubt, that God was the only reason Israel would succeed.
Soon, 32,000 soldiers became 10,000. But that was still too many. Again the army was paired down—with only 300 men remaining to fight.
Each time Gideon was asked to send more troops home, he complied—putting his faith in the Eternal rather than physical warriors.
God spoke to Gideon again: "Get ready to attack, for I have delivered the Midianites into your hands."
Believing God would bring victory to the Israelites, Gideon knew he still had to do his part. He armed the 300 men with clay pitchers, torches and trumpets, later to be used to awaken the Midianites in frenzied confusion. And they set out in the dark of night to claim the victory God had promised.
Gideon was able to be used by God because he had a humble attitude and did not think too highly of himself.
To read more on how God can use a person with a meek and contrite attitude, and what God can accomplish through them, read the entire account of Gideon's life in Judges 6-8.


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While kung fu buffs may be excited for the RZA's upcoming directorial debut Man with the Iron Fists, Hip Hop heads are likely most stoked for the flick's feature-heavy soundtrack. Now, in a recent interview with MTV, the Abbott of the Wu-Tang Clan previews Kanye West's contribution to the soundtrack.
RZA rapped the hook from Yeezy's song on the soundtrack, titled "White Dress," live from the red carpet of the 2012 BET Hip Hop Awards. He said that apart from the hook, the song finds Kanye going in for nearly 40 straight bars, and compared the Chicago emcee's performance to one of Ghostface Killah's verses.
"Oh man, Kanye went in," he said. "He's got a song called 'White Dress,' and the hook goes like, 'I seen you in the club in a tight dress/But I pictured you in a church in a white dress.' That's basically talking about some of the girls, you know, you wanna make a wife out of them, but they gotta get it together. Kanye's a dope lyricist, but what he did on this particular song - it ain't 16 bars; he's going like how Ghostface goes, like 40 bars. Just going, he's going, yo."
Check out the full interview below.
Get More: Music News
RELATED: RZA "The Man With The Iron Fists" Soundtrack Tracklist & Production Credits


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There will be more vocal performances than ever before and an expanded "Music in the Mansion" series during Strathmore's 2013-2014 season, which the North Bethesda concert hall announced this week.
Tickets for the season will go on sale Thursday, July 18. There is again a large range of performances, including Broadway singers, one-woman plays, jazz vocalists and sessions with authors and writers.
Strathmore (5301 Tuckerman Lane) will also have 30 performances in its Mansion concert series, the largest the event has been.
The season will be "anchored by the distinguished and virtuosic voices of women at the pinnacle of their craft — from stage to film, television and literature," according to a press release.
For schedules, information and tickets, visit Strathmore's website.
Flickr photo by Bill in DC
Memorial Day weekend events from around the nation, and the D.C. area, captured in photos.


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The Greek Fisherman is in the heart of Cape Town's V & A Waterfront. Diners are treated to traditional Greek and Mediterranean food and hospitality. Encouraged to sit back and relax while taking in the out-of-this-world view of Table Mountain and the bustle of the working harbour, patrons are treated to great friendly service and delicious food.
Best known for its traditional Greek and Mediterranean menu, the restaurant as added a modern twist to some dishes. With a selection wonderful homemade breads, fresh out of the oven, and over twenty five meze and tapas starters to choose from, including Souvlaki (meat skewers served with tzatziki), Spanokopita (spinach and feta parcels) and Atherina (white bait), diners can feast and soak up the lively atmosphere.
The main courses are predominantly fresh fish and seafood dishes, as well as the highest quality meats, all cooked to perfection over and open charcoal fire for an authentic taste sensation. The wine list has something for everyone.
With the motto "food is family, family is life, life is everything" a meal at the Greek Fisherman is a treat to share with the whole family, and one that won't be forgotten.
This unique African restaurant boasts a circular open fire with Masaai spears that holds a variety of 15 different types of meat, gently roasting away as you enjoy a memorable evening out with family and friends. Meats include game varieties such as crocodile, zebra, giraffe, impala, ostrich, and many more. Six assorted salads and six sauces complement your meal. 'Pap and sauce' and baked potatoes are also offered.
*Distances are shown as the crow flies and are not necessarily the actual travelling
Enjoy delicious and tasty cuisine and a feast for the senses dining on seafood, grills and typical Portuguese dishes.
With attention to detail and personal attention, diners relax in tranquil surrounds dining on mouth-watering dishes such as grilled chicken basted in finger linking peri-peri the Mozambican way, or voluptuous grilled prawns, tasty pregos, trinchados, espetadas and popular feijoada.
Adeguita Portuguese Restaurant tantalises with mouth-watering tastes and flavours the true Portuguese way in a colonial setting that invites relaxation.
An unpretentious family-run restaurant serving superb Portuguese cuisine, A Churrasqueira boasts an exciting menu with a wide selection of choices.
An impressive wine list enhances the experience, with a brilliant selection of both South African and Portuguese wines.
The interior is simple and warm, the service excellent and the prices reasonable. All in all, a wonderful place to dine in comfort.
The delicious menu brims with high quality seafood such as voluptuous Mozambican prawns and a variety of succulent steaks, poultry and traditional dishes.
The warm and inviting ambiance sets the tone for a unique Portuguese experience housed in a luxurious house in Kensington.
Suid Afrika Restaurants,
SouthAfrica Restaurants,


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Fedora from Etsy Follow
Featured in The Anatomy of a Mad Men Wedding.
mad men vintage wedding inspiration.
Real wedding photos by Sarah Q Photography; Fedora, Collar Bar, Skinny Tie
Find Tuxes and Suits Near You:
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Plan, book and shop for your wedding with inspiration as your guide. Plus, save photos to your favorites!


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[ame=http://www.youtube.com/watch?v=DMG_fTyuc-4]YouTube - Video: Arnaud focused on bowl win[/ame]
Twitter: @ChrisMWilliams
Facebook: Cyclone Fanatic
I hope so...
good for him
Originally Posted by CycloneErik
And I'm thinking that cycofan just made the air, baby!
Originally Posted by rworkman83
I'll be honest, Cycofan1 sounds much older than what he makes me believe on here.
Originally Posted by Cycofan1
good for him
Im still trying to figure out if this is a sarcasm comment or not.
"Winning isn't everything, it's the only thing."
During the news, it was noted that he did in fact have a broken hand this season. Not sure if this has been brought up here on CF or not. This time off should have helped.
The grass always looks greener, on the other side of the fence, but the dog over there might be meaner, on the other side of the fence
Stay in your own yard, play in your own yard!
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