The generalization of ratio-cum product estimator for the arbitrary sample design

The generalization of a dual to ratio-cum-product estimator of the finite population total for the arbitrary sample design is given. The approximate variance is calculated and variance estimator is constructed. Some simulation results are presented for the comparison of ratio-cum-product estimator with some known estimators.


Introduction
Many estimators of the finite population parameters are constructed using known auxiliary variables. The classical well known ratio estimator is one of them. Various improvements of this ratio estimator have been considered by many authors. Some estimators use two auxiliary variables. Some other group of estimators are composite. They are constructed by taking a weighted sum of Horvitz-Thompson estimator and some ratio estimator. The sum of corresponding weights need not equal to one. The first of estimators considered in the paper is product, ratio-cum-product (Singh (1965)) and dual to ratio-cum-product estimators [5]. The product estimators are used in case negatively correlated auxiliary variable is available. This estimator behaves similarly as simple ratio estimator. In case two auxiliary variables are known, the ratio-cumproduct estimator may be used. In the paper [5] the dual to ratio-cum-product estimator is considered for simple random sample without replacement. It seems this estimator is more effective compare to the ratio-cum-product estimator. In this paper the dual variables for the construction of dual to ratio-cum-product estimator, are defined for the case of stratified simple random sample and arbitrary sample design. The approximate variance of the dual to ratio-cum-product estimator is calculated and variance estimator is presented.

Main notation, ratio-cum-product estimator
Consider a finite population U = (u 1 , u 2 , . . . , u N ) of N units. Assume three variables y, x, and z are defined on the population U, taking values {y 1 , . . . , y N }, {x 1 , . . . , x N }, and {z 1 , . . . , z N }. We consider y as unknown study variable, x and z by known auxiliary variables, possibly correlated with the variable y. We are interested in the estimation of the finite population total Let simple random sample s ⊂ U of size n is drawn from the population U. Denote The valuest x ,t z , µ x , µ zx ,z, are defined analogously. The ratio, product, and ratio-cum-product estimators are defined as followŝ

Dual to ratio-cum-product estimator
The dual variable for the improvement of ratio-type estimator was used by Bandyopadhyay (1980) and Srivenkataramana (1980). An estimator with two auxiliary variables were considered by Singh et al. [5]. Consider again simple random sampling of size n and introduce the linear transformation of the variables x and z: . Then Correlation coefficient between variables y and x * , Corr(y, x * ) = −Corr(y, x) = −ρ yx and Corr(y, z * ) = −ρ yz . The dual to ratio-cum-product estimator suggested in the paper of Singh et al. [5] iŝ

Stratified simple random sample case
Assume the population U consists of H strata: . . , H , and define two transformations of the auxiliary variable x: The transformations for the variable z are defined analogously. Note that N k=1 x * k (j )= N k=1 x k = t x , j = 1, 2. The relation Corr(y, x * ) = −ρ xy is not valid in the case of stratified sample. The covariances are given by The dual estimators for the both transformations coincide: The dual to ratio-cum-product estimator bê The approximate variance of this estimator can be find by the usual linearization technique. We can expect this estimators be more efficient compare to the corresponding ratio-cum-product estimator.
In the case of the general unequal probability sampling design with the inclusion probability π k of the element k, the dual variable is defined as Using the notationt * one can use the dual to ratio-cum-product estimator for general sample design The approximate variance of the estimatort * RP can be derived using standard linearization technique AV ar(t * RP ) = k,l∈U (π kl − π k π l ) y k − R x g k x k + R z g k z k π k y l − R x g l x l + R z g l z l π l .
Here π kl is an inclusion probability of the pair of elements u k , u l into the sample s. As the estimator of the variance of dual to ratio-cum-product estimator we can use the following estimator V ar (t * RP ) = k,l∈s π kl − π k π l π kl y k −R x g k x k +R z g k z k π k y l −R x g l x l +R z g l z l π l . Here

Simulation study
In this section some empirical study is presented to observe the behavior of the estimators in the case of stratified simple random sample design. A real populations from some Lithuanian Enterprise survey were used for the simulation. During the simulation study several populations were examined. It should be noted that both auxiliary variables initially are positively correlated with the study variable. So, first of all we transform the variable z to dual, and consider the transformed variable as given negatively correlated auxiliary variable. Simulation results presented below show that ratio-cum-product estimator may perform efficiently.
These populations are stratified into three and two strata respectively by the size of the variable x. In the first population 4000 samples and in the second population 2000 samples were drawn. The estimated relative bias (the average bias divided by the true total t y ), average estimate of the variance, approximate true variance and coefficient of variation are calculated. Tables 1 and 2 show that for stratified simple random sampling design dual ratiocum-product estimator can be more efficient than other estimators considered.
For Population II (Table 2), dual ratio-cum-product estimator has a little bit bigger coefficient of variation then ratio-cum-product estimator.