Improved Exponential Estimator in Stratified Random Sampling

In this article we have considered the problem of estimating the population mean  in the stratified random sampling using the information of an auxiliary variable x which is correlated with y and suggested improved exponential ratio estimators in the stratified random sampling. The mean square error (MSE) equations for the proposed estimators have been derived and it is shown that the proposed estimators under optimum condition performs better than estimators suggested by Singh et al. (2008). Theoretical and empirical findings are encouraging and support the soundness of the proposed estimators for mean estimation.


Introduction
In survey sampling, it is well established that the use of auxiliary information results in substantial gain in efficiency over the estimators which do not use such information.However, in planning surveys, the stratified sampling has often proved needful in improving the precision of estimates over simple random sampling.In some cases, in addition to mean of auxiliary variable, various parameters related to auxiliary variable such as coefficient of variation, kurtosis, correlation coefficient, etc. may also be known.For these cases, many authors, such as Upadhayaya and Singh (1999), Sisodia and Dwivedi (1981), Singh and Tailor (2003), Singh et al. (2007), developed various estimators to improve the ratio estimators in the simple random sampling.Kadilar and Cingi (2003) adapted the estimators in Upadhyaya and Singh (1999) to the stratified random sampling.
In this study, under stratified random sampling without replacement scheme, we suggest two improved exponential estimators which are more efficient than estimators proposed by Singh et al. (2008).There are some recent studies proposing the family of estimators without using the exponential function in literature, such as Koyuncu and Kadilar (2009(a,b)), Khoshnevisan et al. (2007), Singh andVishwakarma (2006, 2008) etc.However, in this article, the proposed family of estimators depends on the exponential function.
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 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 the i-th unit in the h-th stratum and h Y and h X denote stratum means.
When the population mean of the auxiliary variable, X , is known, The MSE of t 1 to a first degree of approximation is given by where is the population ratio,

S
is the population variance of a study variable, 2 xh S is the population variance of the auxiliary variable and yxh S is the population covariance between study and auxiliary variables in the stratum h.

Kadilar and Cingi estimator
Kadilar and Cingi (2003) introduced an estimator for population mean using known value of some population parameters in stratified random sampling given by where and a h , b h are the functions of the known parameters of the auxiliary variable such as coefficient of variation C xh , coefficient of kurtosis ) x ( h 2 etc., or some constants in the h th stratum .
The MSE of the estimator t 2 is given by The bias and MSE of the estimators t 4i ( i=0,1,2,3,4) can be obtained respectively by following expressions: We would like to remark that for various values of parameters in (2.4), we get five estimators (for I = 0, 1, 2, 3, 4) as shown in Table 1.The MSE expressions for these estimators are also given in the Table 1.

The values of a h ,b h Estimator MSE of the Estimator
Remark 1 -Here we would like to mention that the choice of the estimator depends on the availability and values of the various parameter(s) used (for choice of the parameters a h and b h refer to Singh et al. (2008) and Singh and Kumar(2009)).Singh et al. (2008) showed that all of these modified estimators (t 4i ) have a smaller MSE than the MSE of the stratified version of exponential ratio type estimator t 3st under certain conditions: Note that t 3st is the same estimator with t 40 .
Singh et al. ( 2008) reported the minimum value of mean square error of the estimator t 4i as where, c is combined correlation coefficient in stratified sampling across all strata.It is calculated as

Suggested modified exponential ratio estimator
In stratified random sampling, Kadilar and Cingi (2005) introduced a ratio estimator as ) where the constant k is obtained by minimizing the MSE of t 5 .
Motivated by Kadilar and Cingi (2005), we propose the following modified estimator To obtain the MSE of t 6i to the first degree of approximation, let us define . Using these notations we have where Expressing (3.2) in terms of e's, we have where MSE of the estimator t 6i can be rewritten as To obtain the optimum value of k, we partially differentiate the expression (3.6) with respect to k and put it equal to zero as follows: The bias of the estimator t 7i , to the first order of approximation, is obtained as From equation (4.4), we observe that to the first order of approximation the proposed estimator is biased.From (4.3), squaring and than retaining the terms of e's upto power two, we have where and In order to find optimum value of estimator t 7i , we differentiate MSE (t 7i ) with respect to i 1 and . i The optimum values of i 1 and i 2 are Remark 3 -Here we would like to mention that the optimum values of i 1 and i 2 require information about population parameters.However, in applications, to have this information is sometimes impossible.However in repeated surveys or studies based on multiphase sampling, where information is gathered on several occasions (or based on past experience) it may be possible to guess the value of these parameters quite accurately.Even though this approach may reduce the precision, it may be satisfactory provided the relative decrease in precision is marginal.(For further details refer to Sukhatme et.al. ((1984), pp.260), Singh and Vishwakarma (2006) and Koyuncu and Kadilar (2009(b)).

Efficiency comparisons
For the estimators, efficiency comparisons are given below.
In this section, we first compare modified estimator t 6i given in (3.2), with the estimator t 4i given in (2.4).
As the condition (5.1) is always satisfied, we can say that the suggested estimator is more efficient than exponential ratio estimator in stratified random sampling in all conditions.Note that when i = 0, MSE (t 40 ) = MSE (t 3st ).
Second, we compare the improved estimator t 7i , given in (4.1), with estimator t 4i , given in (2.4) as follows: . Then the expressions can be obtained on solving as where Next, we compare the improved estimator t 7i given in (4.1) with the estimator t 6i given in (3.2).MSE (t 7i ) < MSE (t 6i ), i = 0, 1, 2, 3, 4, where Finally, we compare the improved estimator t 7i , given in (4.

Conclusion
As shown in theory, we also observe from Table 3 that suggested estimators t 6i (including optimal value) performs better than corresponding estimators without optimal values, t 4i .Besides, we see that improved estimator t 7i and its members are always more efficient than corresponding estimators (without optimal value of k i and with * i k ) for this data set.
respectively, we get the minimum MSE of the estimator t 7i .

Table 2 : Data Statistics
Kadilar and Cingi (2003)use the data set earlier presented byKadilar and Cingi (2003).In this data set, Y is the apple production amount and X is the number of apple trees in 854 villages of Turkey in 1999.The population information about this data set is given in Table2.

Table 3 : MSE Values of Estimators
In Table3we have reported the bias and MSE values of the different estimators.Here we would like to mention that the MSE of the different estimators depend on choice of a h and b h .The minimum MSE attained bySingh et al. (2008)estimator was 218374.8898.From Table3we observe that the estimator t 7i (i=0,1,2,3,4) under optimum conditions performs better than all other estimators proposed bySingh et al. (2008).We also observe from the table that the estimator t 73 is having larger MSE than the minimum MSE attained bySingh et al. (2008).So, for this data set choice of a h = 2h (x) and b h =C xh is not a good choice.Also, for the choice a h =C xh and b h = the MSEs of the estimators t 44 , t 64 and t 74 is minimum.