A family of estimators of population mean using multi-auxiliary variate and post-stratification

Abstract. This paper suggests a family of estimators of population mea n using multiauxiliary variate based on post-stratified sampling and its properties are studied under large sample approximation. Asymptotically optimum estim ator in the class is identified alongwith its approximate variance formulae. The proposed class of estimators is also compared with corresponding unstratified class of esti mators based on estimated optimum value. At the end, an empirical study has been carrie d out to support the proposed methodology.


Introduction
Stratification is one of the most widely used techniques in sample survey design serving the dual purpose of providing samples that are representative of major sub-groups of the population and improving the precision of estimators [1].Stratified sampling presupposes the knowledge of strata size as well as the availability of a frame for drawing a sample in each stratum [2].However application of this technique presupposes the knowledge of strata size and the availability of sampling frames within strata.In many socio-economic and agricultural surveys where it is necessary to partition the finite population under consideration, due to its heterogeneity, into different sub-populations (strata), the sampling frame within strata may not be available.However frame for entire population may be available and percentage of population units falling into different strata may be known.Under such circumstances usual stratified sampling can not be used and thus an effort is made to get over the problem through post-stratification which consists in selecting a sample from the whole population by the procedure of simple random sampling without replacement followed by the classification of the selected sample units by strata and then treating it as if it were stratified sample, for instance, see [1,[3][4][5][6][7][8][9][10][11].
It is further noted that in sample surveys, the information on an auxiliary variate correlated with the principal (study) variate under study is either readily available or may be made available by diverting a part of the survey resources.This information may be utilized to increase the precision of estimators of population mean Y of the study variate y.Such an information is the known population mean X of the auxiliary variate x.For illustration, the average farm size in a local government area or district may be known while the problem is to estimate the average area under a particular crop per farm.The strata may be formed according to farm size, the percentage of farms falling into different size groups may be known but the identity of farms within a size group may not be known, see [12].
We assume that the population comprises N units, which can be uniquely partitioned into L strata of size N 1 , N 2 , . . ., N L such that L h=1 N h = N .The strata weights W h = N h /N (h = 1, 2, . . ., L) are assumed known.Let (y hi , x hi ) (i = 1, 2, . . ., N h ) denote the values of variates (y, x) respectively for i-th unit in h-th stratum and Y h and X h denote strata means.A simple random sample of size n is drawn without replacement from the population which results into the configuration n = (n 1 , n 2 , . . ., n L ), n h denoting the number of units in the sample falling in stratum h, L h=1 n h = n.Assume that n is large enough so that the probability of n h being zero is small (i.e.P r(n h = 0) = 0).Based on the foregoing procedure which is known as post-stratification, the usual unbiased post-stratified estimators for population means x hi are the means of the n h sample units that fall into the h-th stratum whose size N h is assumed to be known.
For given configuration of sample n = (n 1 , n 2 , . . ., n L ) we have see [1], where Using the results from [13] for E(n −1 h ), to the terms of order n −2 , we have where f = n/N is over all sampling fraction.It is known that when the auxiliary information is used at the estimation stage, the ratio estimator is the best among a wide class of estimators when the relation between y and x, the variate under study and the auxiliary variate respectively, is a straight line through the origin and the variance of y about this line is proportional to x, see [14].In such a situation the ratio estimator is as good as regression estimator.In many practical situations, the regression line does not pass through the origin.In these situations, the ratio estimator does not perform equally well as that of regression estimator.Keeping this fact in view and also due to the stronger intuitive appeal statisticians are more inclined towards the use of the ratio and the product estimators and hence a large amount of work has been carried out towards the modification of ratio and product estimators, for instance, see [11,[15][16][17] etc.These authors have proposed various estimators under simple random sampling without replacement (SRSWOR) and stratified random technique which under some realistic conditions is more efficient than the mean per unit estimator, the ratio and the product estimator are efficient as the linear regression estimator in optimum case.It is to be mentioned that the problem of estimation of population mean Y of the study variate y based on post-stratification and auxiliary information has not attracted much attention of survey statisticians, for instance, [12] and [18].
In this paper, following approaches developed by [19] and [20], we have suggested a family of estimators of population mean Ȳ of the study variate y based on poststratification using multi-auxiliary variate and its properties are studied.
Defining ε 0 = (y P S −Y ), ε k = (x kP S −X k ) and ε T = (ε 1 , ε 2 , . . .ε p ), we have for a given configuration of n = (n 1 , n 2 , . . ., n L ), the values of the conditional expectations: and if n h is large, to terms of order n −1 h , the conditional expected values are where Putting the above results in matrix notations, we have where The unconditional expectations are: and for large n, to terms of order n −1 , the unconditional expected values are given by Putting the above results in matrix notation, we have where

The suggested family of estimators
Let X T = (X 1 , X 2 , . . ., X p ) denote the row vector of p elements X 1 , X 2 , . . ., X p .
Whatever be the sample chosen let (y P S , x T P S ) assume values in a closed convex subset, Q, of the (p + 1) dimensional real space containing the point (Y , X T ).We suggest a family of post-stratified estimators for the population mean using multi-auxiliary variable as: where G(y P S , x T P S ) is a function of y P S , x 1P S , x 2P S , . . ., x pP S such that and such that it satisfies the following conditions: 1.The function G(y P S , x T P S ) is continuous and bounded in Q, 2. The first and second order partial derivatives of the function G(y P S , x T P S ) exist and are continuous and bounded in Q.
Expanding the function G(y P S , x T P S ) about the point (Y , X T ) in a second order Taylor's series, we obtain: where p ) with G (1) k = (∂G(y P S , x P S )/∂x kP S )| (Y ,X k ) and G (2) denotes the p × p matrix of the second partial derivatives of G(•) with respect to x P S about the point (Y , X T ).Expressing (7) in terms of ε's and noting that G(Y , X T ) = Y T , we have Taking conditional expectation in (8) and noting that second derivatives are bounded.
Thus we arrived at the following theorem: From Theorem 1, it follows that the bias of the estimator Y G is of the order n −1 h , and hence its contribution to the mean squared error of Y G will be of the order of n −2 h .Now we prove the following result: Theorem 2. Up to terms of order n −1 h , the conditional variance of Y G is minimized for and the conditional minimum variance is given by Proof.From (8), we have upto terms of order n −1 h , which implies that ∂G(•) which is minimized for where . Thus the resulting conditional variance is given by where and R is the multiple correlation coefficient between y P S and the vector x P S .Hence proved the Theorem 2.
The conditional variance of any estimator of the class (5) can be obtained from (11).From (11) the conditional minimum variance (i.e.min.Var( Y G |n)) is not larger than the conditional variance of the unbiased estimator y P S , since Taking unconditional expectation in (8) and noting that second derivatives are bounded, we have: Theorem 3 shows that the bias of the estimator Y G is of the order n −1 , and hence its contribution to the mean square error (MSE) of Y G will be of the order n −2 .Thus, to the first order of approximation, the unconditional variance of Y G will be the same.

Theorem 4. Upto terms of order
and the unconditional minimum variance of Y G is given by where S * * 2 Proof.From (8), we have upto terms of order n −1 , which implies that ∂G(•) Using the results (3) and (4) in the above expression we get the unconditional variance over all possible distribution, for large n, to the terms of order n −1 , as: which is minimized for where where and R * is the multiple correlation coefficient between y P S and the vector x P S .Hence proved the Theorem 4.
The unconditional variance of any estimator of the class (5) can be obtained from (16).From ( 16), the min.Var( Y G ) is not large than the unconditional variance of the unbiased estimator y P S , since A * T D * −1 A * > 0.
Let G (1) Y , X = −αD * −1 A * , is a departure from the optimum value (α > 0 is a constant), we have It is well known that the unconditional variance of the usual unbiased estimator y P S is Thus for any G (1) (Y , X T ), we find from ( 18) and ( 19) that which shows that the proposed class of estimators Y G would be better than usual unbiased estimator y P S as for as 0 < α < 2.
Remark 1.It is to be mentioned that optimum estimators in the class are not unique but all of them have the same variance given either by (13) or (18).We also note that in practice the value of δ 0 = −D −1 A at (12) or G (1) 0 = −D * −1 A * = δ * 0 at (17) may not be known.However, they can be estimated by where In such a case we may define a class of estimators based on estimated optimum value δ 0 as: where G * y P S , x T P S , δ T 0 ) is a function of (y P S , x T P S , δ T 0 ) such that: Under (24) the class of estimators Y * G at (23) is expected to have, to the first order of approximation, the conditional and unconditional variances respectively as and

Comparison with corresponding unstratified multivariate estimators
We assume that information on p auxiliary variates x 1 , x 2 , . . ., x p is available.A simple random sample of size n is drawn from the given finite population of size N .Let y i and x i denote the values of the variates y and x k of the i-th unit of the sample, k = 1, 2, . . ., p; i = 1, 2, . . ., n. Defining: Further ρ 0k and ρ kl denote the correlation coefficients between the variates y and x k and between the x k and x l . Define where Putting the above results in matrix notations, we have Let X T = (X 1 , X 2 , . . ., X p ) denote the row vector of p elements X 1 , X 2 , . . ., X p .
Whatever be the sample chosen, let (y, x T ) assume values in a closed convex subset, W , of the (p+1) dimensional real space containing the point (Y , X T ).Following [21] one may define a class of estimator of population mean Y as where G(y, x T ) is a function of y, x 1 , x 2 , . . ., x p such that G Y , X T = Y , for all Y and such that it satisfies the following conditions: 1.The function G(y, x T ) is continuous and bounded in W .
2. The first and second order partial derivatives of the function G(y, x T ) exist and are continuous and bounded in W .
To the first degree of approximation, the variance of Y G is given by Var Y (1) which is minimized when where G (1) (Y , X T ) denotes the p elements column vector of the partial derivatives of G(y, x T ) with respect to x T about the point (Y , X T ).
Thus the resulting minimum variance of Y G is given by where and R * * is the multiple correlation coefficient between y and (x 1 , x 2 , . . ., x p ).
Following [18], and from ( 18) and ( 23 Remark 2. In practice, the exact optimum value η 0 of G (1) (Y , X T ) at (29) is not known, it is available to replace it by its consistent estimate of η 0 from the sample data at hand.Thus following the procedure outlined in [21], we define a class of estimators for population mean Y (based on estimated optimum values) as: where Under the condition (32), it can be shown to the first degree of approximation that the variance of Y (1) sample size n = 65.4 (can be rounded to 65) was selected from the entire population of N = 654 persons.Out of 65.4 persons, 27.9 persons were found to be from stratum-1, 3.9 persons were from stratum-2, 31.0 persons were from stratum-3 and 2.6 persons were from stratum-4.We used R-code given in the Appendix to produce the results shown in Table 3.For T = 0.2, the values of the population correlation coefficients between F EV and Age are 0.76839, −0.05719, 0.82536 and 0.42073 in the first, second, third and fourth post-stratum, respectively.The values of the population correlation coefficients between F EV and Height are 0.85847, 0.24024, 0.90837 and 0.78591 in the first, second, third and fourth stratum, respectively.In the same way, the values of the populations correlation coefficients between Age and Height are 0.77642, −0.09219, 0.84230 and 0.35209 in the first, second, third and fourth stratum respectively.In this particular situation, the percent relative efficiency of the post-stratified sampling estimator θ 1 with respect to the simple sample mean estimator θ 0 remains 558.13 % and that of the non-stratified estimator θ 2 remains 514.11 %.In the same way, the results in Table 3 are readable for other values of T .It is to be noted that so long as the value of T is less than or equal to 2.0, the percent relative efficiency of the post-stratified estimator remains around 557 % and that of the non-stratified estimator remains around 511 %.As soon as the value of T becomes 2.3, the relative efficiency of the post-stratified estimator drastically reduces to 296.36 % and that of non-stratified estimator reduces to 247.49 %.For higher value of T equal to 2.5, the relative efficiency of the post-stratified sampling estimator reduces to 105.93 % and that of non-stratified sampling estimator reduces to 103.92 %.Thus, we conclude that the proposed post stratified sampling estimator can be used to estimate population mean of a study variable in the presence of multi-auxiliary variables more efficiently than a non-stratified sampling estimator.inp10<-read.fwf("c:\\rc\\out01",c(14,10,8,5,5),header=FALSE,sep="\t",as.is=FALSE,skip=0,col.names=names); inp01<-read.fwf("c:\\rc\\out10",c(14,10,8,5,5),header=FALSE,sep="\t",as.is=FALSE,skip=0,col.names=names); inp00<-read.fwf("c:\\rc\\out11",c(14,10,8,5,5),header=FALSE,sep="\t",as.is=FALSE,skip=0,col.names=names); y1<-c(inp11 the study variate y and the auxiliary variate x are y P S = L h=1 W h y h and x P S = L h=1 W h x h , where ) it can be shown that the proposed class of estimators Y G in post-stratified sampling in unconditionally more efficient than the corresponding unstratified class of estimators Y (1) G .

Table 1 .
Descriptive parameters of F EV , Age and HT .

Table 2 .
Pearson correlation coefficient values for four strata.

Table 3 .
Relative efficiency of the post-stratified and non-stratified estimators with respect to the sample mean estimator.