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Modified estimators of finite population distribution function based on dual use of auxiliary information under stratified random sampling

In survey sampling, information on auxiliary variables related to the main variable is often available in many practical problems. Since the mid-twentieth century, researchers have taken a keen interest in the use of auxiliary information due to its usefulness in estimation methods. The current study presents two new estimators for the distribution function of a finite population based on dual auxiliary variables. The new estimators can be used in situations where the researchers face some sort of complex data set. The mathematical equations for the bias and mean square error have been obtained for each proposed estimator. Besides, an empirical study simulation study has also been conducted to analyse the performance of estimators. It is found that the new suggested estimators of the distribution function of a finite population are more accurate than some of the existing estimators.


Introduction
Many researchers have studied the use of auxiliary variables in the literature of survey sampling to increase the efficiency of their developed estimators for estimating common parameters like mean, median, variance, and standard deviation. Traditional ratio, product, and regression estimators provide efficient results for unknown parameters in such circumstances.
Out of many practices, the ratio method and product method has been widely used for estimating unknown population parameters, when there is a high positive and a high negative correlation between study variable and auxiliary variable. In the past, several authors introduced many ratio type and product type estimators by using different type of linear transformation of original auxiliary variables. The drawback of these class of estimators are that it uses a very specific linear transformation of auxiliary variable that restrict the scope of applications of this class, in practice. To overcome this drawback, we here propose a generalized class of ratio in regression type exponential estimators of population distribution function under a very general linear transformation of auxiliary variable.
Simple random sampling works quite well if the population of interest is homogeneous. When the population of interest is heterogeneous, however, it is preferable to apply stratified random sampling rather than simple random sampling. In stratified random sampling, we divide the entire aggregate into numerous non overlapping groups or subgroups called strata. These groupings are completely homogeneous, and a sample is taken from each stratum separately. The values of the Nh must be known in order to get the most out of stratification. After the strata have been determined, a sample is taken from each stratum, and the drawings are done separately. The entire technique is represented as stratified sampling if a simple random sample is collected from each stratum. To divide the sample into strata, different researchers utilized different sample allocation procedures. If the sample size in each stratum is large enough, using a distinct ratio estimate in each stratum is more precise. As a result, we'll apply the proportional allocation strategy in this article. The population mean under stratified random sampling has received more attention. Stratification enhances efficiency, when the variance between strata is substantially greater than the variance within strata,. The problem of measuring the function of finite population cumulative distribution (CDF) arises when interest lies in knowing the proportion of study variables that are below or equal to a certain value. The need for CDF in many situations is greater than ever. For example, a physician could be interested in knowing what percentage of the population consumes 35% or more of their calories from trans fats. A soil scientist, for example, could be interested in determining the clay percentage distribution in the soil. In addition, policymakers may be curious about the percentage of people living in a developing country who are poor. The CDF has been computed using information on one or more auxiliary variables in survey sample literature. Chambers and Dunstan, 1 Rao et al., 2 Rao, 3 Kuk,4 Ahmed and Abu-Dayyeh, 5 Rueda et al., 6 Singh et al., 7 Hussain et al., 8 and Hussain et al. 9 proposed two new estimators for estimating the finite population distribution function using supplementary information using simple and stratified random sampling schemes. In practice, there needs to be more research on the use of both auxiliary variables and the finite population distribution function.
In survey sampling literature, the authors have estimated finite population distribution function (CDF) using on one or more auxiliary variable. Dual use of auxiliary variable has been rarely attempted while estimating finite population distribution function, therefore we motivated towards it. In this article we proposed two new estimators which are competing the existing estimators and estimators proposed by Hussain et al. 9 The paper offers two new estimators for estimating finite population distribution functions under stratified random sampling that leverage dual usage of auxiliary information. The bias and mean square error of the proposed estimators have been expressed up to first order of approximation. Cochran, 10 Murthy, 11 Bahl and Tuteja, 12 Rao , 13 Singh and Kumar, 14 Grover and Kaur, 15 and Hussain et al. 9 have all demonstrated that the proposed estimators are more efficient than traditional unbiased estimators, both theoretically and empirically.
The problem of estimating the finite population CDF arises when the interest lies in knowing the proportion of values of the study variable that are less or equal to a certain value. There are situations where estimating the CDF is deemed necessary. For example, for a nutritionist, it is interesting to know the proportion of population that consumes 25% or more of the calorie intake from saturated fat. Similarly, a soil scientist may be interested in estimating the distribution of clay percent in the soil.
In addition, policymakers may be interested in knowing the proportion of people living in a developing country below the poverty line.

Sampling design and notations
When the population is heterogeneous, stratified random sampling should be used instead of simple random sample. In stratified random sampling, we split the diverse population into a number of non-overlapping groups or subgroups termed strata. These groupings are completely homogeneous, and a sample is taken from each stratum separately. To disperse the samples in the strata, surveyors employ a variety of sample allocation techniques. If the sample size in each stratum is big enough, using independent ratio estimates in each stratum is more precise. As a result, we will apply the proportional allocation method in this article. Many publications offered many ratio type estimators in stratified sampling by changing the auxiliary variable, such as Kadilar and Cingi, 16 Kadilar and Cingi, 17 Koyuncu and Kadilar, 18 Shabbir and Gupta, 19 Aladag and Cingi, 20 Malik and Singh. 21 Let Ω = {1, 2, . . . , N} be a finite population of N units, which is divided into L homogeneous strata, where the siRe of h th stratum is N h , for h = 1, 2, . . . , L, in such manner L h=1 N h = N. Assume that Y and X be the study y h and auxiliary variable x h , where i = 1, 2, . . . , N h and h = 1, 2, . . . , L, a sample n h is drawn in such a manner L h=1 n h = n, where n is the sample siRe.
be the population and sample distribution functions of Y and X under stratified random sampling, respectively, where W h R h be the population and sample means of X and Z under stratified random sampling, respectively, where, To find the properties of the existing and proposed estimators of F(ty), we consider the following relative error terms under stratified random sampling. Let such that E(ν i ) = 0 for i = 1, 2, 3, 4, where E(·) is the mathematical expectation of (·). Let

Existing estimators
Several approximations of the finite population distribution function under stratified random sampling are described in this section. In all these estimators (Existing) the authors use single auxiliary information (Variable), except Hussain et al. 9 Under the first order of approximation, the biases and MSEs of these adapted estimators are calculated.

Mean estimator
The traditional unbiased mean estimator F(ty), is given bŷ The variance ofF * SRS st (ty), is given by Cochran 10 In stratified random sampling, the usual ratio estimator F(ty) is provided bŷ The bias and MSE ofF * R (ty), are given by If R F ty F tx > C F tx /(2C F ty ), thenF * R (ty) is better thanF * SRS (ty) in terms of MSE. Murthy 11 Suggested the usual product estimator F(ty) in stratified random sampling, is given bŷ The bias and MSE ofF * P (ty), are given by Bias(F * P (ty)) = F(ty)μ 1100 , and MSE(F * P (ty)) ≅ F 2 (ty)(μ 2000 + μ 0200 + 2μ 1100 ).

Regression estimator
The usual regression estimatorF Reg (ty), in stratified random sampling, is given bŷ where k is an appropriate chosen constant. The minimum variance ofF * Reg (ty) at the optimum value k (opt) = (F(ty)μ 1100 )/(F(tx)μ 0200 ) is Here (12) may be written as Rao 13 Suggested an improved difference-type estimator F(ty), in stratified random sampling, is given byF * where k 1 and k 2 are constants that are unknown. To the first order of approximation, the bias and MSE ofF * R,D (ty) are given by The ideal values of k 1 and k 2 , as obtained by minimizing (15), are k1 and k2, respectively .
The minimum MSE ofF * R,D (ty) at the optimum values of k 1 and k 2 is Here (16) may be written as Singh et al. 14 Suggested generalized ratio-type exponential estimator of F(ty), in stratified random sampling, is given bŷ where, a = 1 and b = 0. The bias and MSE ofF * S (ty), to the first order of approximation, are given by where θ = aF(tx) / (aF(tx) + b).

Grover and Kaur 15
Introduced ratio-type exponential estimator of F(ty), in stratified random sampling, is given bŷ where the constants k 3 and k 4 are unknown. To the first order of approximation, the bias and MSE ofF * G,K (ty) are given by The minimum values of k 3 and k 4 are, .
Here (64) may also be written as where It can be seen thatF * H1 (ty) is more precise thanF * Reg (ty).
Here (29) may be written as where It is clear thatF * H2 (ty) is more precise thanF * Reg (ty).

Proposed estimators
The theory of stratified random sampling deals with the characteristics of the estimates with a great choice of sample size n h to get maximum precision. When the correlation exists between the study variable and the auxiliary variable, then there is a possibility that the correlation also exists between the study variable and CDF as well as the rank of the auxiliary variable.
In the literature of survey samping consider here, the authors are used one or more auxiliary varibles (Information) for estimation of finite population distribution function. Using dual auxiliary varibles in the felid of estimation of finite population distribution function are rarely attempted. The principal advantages of our proposed ratio-in-regression exponential type estimators under statrtified random sampling are that it is more flexible and efficient existing then the existing estimators.
Motivated by Hussain et al., 9 we propose a new ratio in a regression type exponential estimator of the finite population distribution function in stratified random sampling, including supplementary information in the form of CDF mean and rank of the auxiliary variable.

First proposed estimator
We use the same idea as the first proposed family of estimators Hussain et al., 9 and estimate finite population CDF, which concerned with CDFs of study and auxiliary variables along with the mean of the auxiliary variable.

Second proposed estimator
Here we use the same idea of second proposed family of estimators of Hussain et al., 9 and estimate finite population CDFs which concern CDFs of study and auxiliary variables along with ranks of the auxiliary variable.
where k 14 , k 15 and k 16 are suitable chosen constants. The estimatorF * Prop 2 (ty), in terms of errors, we havê The optimum values of k 14 , k 15 and k 16 , determined by minimizing (39), are given by The minimum MSE ofF * Prop 2 (ty), at the optimum values of k 14 , k 15 and k 16 is given by where ,

Empirical study in stratified random sampling
In this section, we conduct a numerical study to investigate the performances of the existing and proposed CDF estimators in stratified random sampling. For this purpose, four populations are considered. The summary statistics of these populations are reported in         X: Apple production amount in 1998.
From Table 5 and Table 6, in terms of mean squared error and PRE, it is clear that proposed estimators i.e,F * Prop 1 (ty) andF * Prop 2 (ty) performs better than the estimatorŝ F * SRS st (ty),F * R (ty),F * P (ty),F * BT,R (ty),F * BT,P (ty),F * Reg (ty),F * R,D (ty),F * S (ty),F * G,K (ty), F * H1 (ty), andF * H2 (ty). As we increase the sample size the mean square error values decrease, and percentage relative efficiency gives the best results, which are the expected results.

Simulation study
We have generated two populations of size 1000 from multivariate normal distribution with different covariance matrices. The results of simulation are given in Tables 7 and 8. The population means and covariance matrices, are given below:  We estimate the MSE using k = 1000 samples of diverse sizes selected from each population. Three different sample sizes n = 100, 150, 200 are taken from both populations. Table 7 shows that the proposed estimatorsF * Prop 1 (ty) andF * Prop 2 (ty) performs better as compared to all other existing estimators for both populations in terms of MSEs. We have also seen that as the sample size increases MSE of all the decreases Table 8 shows that the proposed estimatorsF * Prop 1 (ty) andF * Prop 2 (ty) performs better as compared to all other existing estimators for both populations in terms of the PREs.

Conclusion
In this article, we propose ratio-in-regression type exponential estimator for the finite population distribution function under stratified random sampling, which required an ancillary variable on the sample mean and rank of the auxiliary variable. Expressions for mean square error of the proposed estimator are derived up to first order of approximation and comparison is made with the estimators mentioned herein. According to results of real data sets, and simulation it is perceived that the proposed estimator of (F * Prop 1 (ty),F * Prop 2 (ty)) performs better in terms of percentage relative efficiency, than usual estimator of estimator of Hussain et al., 9 Cochran, 10 Murthy, 11 Bahl and Tuteja, 12 regression estimator, Rao, 13 Singh et al., 14 and, Grover and Kaur. 15 A simulation analysis is also carried out to assess the robustness and generalizability of the propose estimator. The simulation study's findings also confirm the utility of the proposed estimator. A numerical study is carried out to support the theoretical results. Therefore, we recommend the use of proposed estimators for efficiently estimating the finite population finite population distribution function under stratified random sampling.