Improved Family of Ratio Estimators of Finite Population Variance in Stratified Random Sampling

22 20 , , , yx k k y x y x λ µ µ =     Biostatistics and Biometrics Open Access Journal ISSN: 2573-2633 Abstract This paper introduces a new family of exponential ratio estimators of population variance in stratified random sampling and studies its properties. Based on Bahl & Tuteja [1], Kadillar & Cingi [2] and Solanki et al. [3], membership of the new family of estimators is identified. Analytical and numerical results show that under certain prescribed conditions, the new estimator has equal optimal efficiency with the


Introduction
In sampling theory population information of the auxiliary variable such as the total, mean and variance are often used to provide additional information which help to increase the efficiency of the estimation of the population parameter(s) of interest. When the information on an auxiliary variable is known, one can use the ratio, product and regression estimators to improve the performance of the estimator of the study variable. When the correlation between the study variable and the auxiliary variable is positive, ratio method of estimation is quite effective. Many authors like [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] among others have proposed different ratio estimators (of total or mean) in sample surveys.
The estimation of the finite population variance has been of great significance in various fields such as Industry, Agriculture, Medical and Biological sciences. Various authors such as [4,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] have used auxiliary information to improve the efficiency of the estimator of population variance of the study variable. In this paper, the problem of constructing efficient estimator of the population variance is considered and a new family of exponential ratio estimators of population variance in stratified random sampling that is as efficient as the general regression estimator is introduced.

Basic notations and definitions
Consider a finite population ( ) 1,2 , , N π π π Π = … of size (N). Let (X)and (Y) denote the auxiliary and study variables taking values X i and Y i respectively on the i-th unit ( 1, 2, , ) i i N π = … of the population. Let the population be divided into K strata with N k units in the k th stratum from which a simple random sample of size n k is taken without replacement. The total population size be respectively. Associated with the ith element of the h th stratum are y ki and x ki with 0 ki x > being the covariate; where y hi is the y value of the ith element in stratum k, and x ki is the x value of the ith element in stratum , 1, 2, ,

The suggested estimator
The suggested exponential ratio estimator of population variance is given by: Where , k k α τ and k η are suitably chosen scalars such that k α and k τ satisfies the condition 1 ; .
Taking the expectation of (2), the Bias of estimator 2 pr S is obtained as: Squaring both sides of (2) and taking expectation, the Mean Square Error is obtained by Taylor's series approximation as: The suggested estimator 2 pr S attains its optimal efficiency if Substituting the value of ,

Alternatively
The suggested estimator 2 pr S would attain its optimal efficiency if: Remark 1: Following from ( (7), (8), (9)), it should be noted here that the proposed estimator 2 pr S would also attain its optimal efficiency with minimum MSE given as in (6), when either of the following optimality conditions is satisfied:

Membership of the proposed estimator
This section studies the properties of the proposed estimator and identifies some special members of its family and derives their mean square errors (MSEs) under certain prescribed conditions.  exponential ratio-type estimator of population variance in stratified random sampling given as:

Efficiency comparisons
This section compares the optimal MSE of the proposed ratio estimator of population variance ( )

Regression estimator
The proposed ratio estimator would be more efficient than the general regression estimator of population variance 2  (20) Taking the expectation of (20), the Mean Square Error is obtained by Taylor's series approximation as: Where the value of B that minimizes (21) is given by opt ψ Β = Ω (22) Substituting the value of opt Β in (22) for Β in (21) (6) and (23), it is evidenced that under certain prescribed optimality conditions, the proposed ratio estimator ( ) 2 pr S has equal optimal efficiency with the general regression estimator (S 2 REG ). The implication is that the proposed exponential ratio estimator of population variance is as good as the general regression estimator of population variance.

Stratified random sampling estimator
The proposed estimator would be more efficient than the stratified random sampling estimator of population variance if:  In this section, the performance of the proposed ratio estimator is assessed with every identified existing estimators of its family and the classical ratio estimator of population variance by Isaki [4]. The merits of the suggested ratio estimator over other existing estimators were judged using the Data Statistics in Table 1.

Discussion of Results
Analytical comparisons showed that the proposed ratio estimator of population variance under certain realistic conditions is more efficient than the unbiased stratified sampling estimator of population variance, classical ratio estimator of population variance by Isaki [4], Bahl & Tuteja [1] ratio-type estimator, Solanki et al [3] ratio-type estimator and Kadillar & Cingi [2] ratio-type estimator but has equal optimal efficiency with the regression estimator of population variance. Numerical results for the percent relative efficiency (PREs) in Table 2 reveals that the proposed estimator ( )  [3] ratio-type estimators of population variance respectively. Also, in using the proposed ratio estimator, one will have 117 percent efficiency over the Kadillar & Cingi [2] ratiotype estimator of population variance. Generally, the proposed ratio estimator fares better than every identified existing estimators of its family and has equal optimal efficiency with the regression estimator of population variance.

Conclusion
Sequel to the discussion of results above, it is concluded that the proposed ratio estimator ( ) 2 , pr S fares better than every identified existing estimators of its family and has equal optimal efficiency with the general regression estimator ( ) 2 REG S which has always been preferred because of its efficiency. Therefore, the new ratio estimator of population variance in stratified random sampling should be preferred in practical situations by survey researchers as it provides consistent and more precise estimates of the population parameters of interest.