Improved Ratio-Cum Regression Estimator Using Two Auxiliary Variables In Single Phase Sampling

  Conclusion

The proposed estimator No Extreme Value (NEV) was asymptotically efficient over the reviewed estimators except in comparison with the regression estimator of ([AK14]) where it was partially efficient.

1. Introduction

  has incorporated the use of more than one estimator. This is what is now known as mixed estimation method in single phase sampling. The use of mixed estimation over the years has been confirmed to enhance the efficiency of any estimator by . Some notable authors that have use this method include  and ,  and . This study is extending the work of ([KS13]) into mixed estimation (Ratio-cum-regression) by combining ([KS13]) improved ratio and regression estimators in the order of ([KC05]). The proposed estimator shall assume that both the study and the auxiliary variables have no extreme value in their distributions. This proposed estimator shall be called NEV. This study shall test the performance of NEV theoretically, empirically and using percentage relative efficiency against the improved ratio estimator of ([KS13]), the improved ratio estimator of ([AK14]) and the improved regression estimator of ([AK14])

1. METHODOLOGY

2.0. All symbol used in this article have been clearly defined in Appendix A.

2.1 Review on ([S72]) correction factor

 ([S72]) has advanced solution to extreme value by introducing a correction constant  such that if there exists extreme large value in a distribution and  is the sample mean using Simple Random Sampling without Replacement (SRSWOR), then  will be subtracted from  to obtain the corrected mean. This is stated as

                                                                                                                                                     (1)   

Likewise, if there exists extreme low value in a distribution and  is the sample mean with SRSWOR, then  will be added to  to obtain the corrected mean. This is stated as

                                                                                                                                       (2)

This can be written in a compressed form as

                                                                     (3)   

 is the correction constant. The minimum variance of  up to first order of approximation is given as

                                                                                                                                   (4)

where    and the optimum value of  is given as

                                                                                                                                              (5)

2.2 Review on ([KC05])

 ([KC05]) has advanced an estimator, which was derived from the combination of the regression estimate of  and the estimator of ([A-D03]). ([KC05]) estimator is given as

                                                                                     (6)

where and were real numbers and ,     . Here,  and   are the sample variance of  and    are the sample covariances between y and   and between y and   respectively.   The MSE is given as:

                                                      (7)

2.3 Review on ([KS13]) ratio estimator

 ([KS13]) has proposed an improved ratio estimator using one auxiliary variable with extreme value.  The estimator is given as

The corresponding MSE is given as:

where   , is the mean square error of the conventional ratio estimator.  

  2.4 Review on ([KS13]), regression estimator

([KS13]) has proposed an improved regression estimator using one auxiliary variable with extreme value.  The estimator is given as               

                                                   =   +  (  - ),                                                                       (10)

with the corresponding MSE as

                                                 V ( )_opt    M ( )   ,                                                        (11)

where M ( )  =       and b is the sample regression coefficient.

2.5 Review on ([AK14]) ratio estimators

([AK14]) has proposed an improved ratio estimator using two auxiliary variables with extreme value. The estimator is given as

The corresponding MSE is presented as

where .

2.6 Review on ([AK14]) regression estimators

([AK14]) has proposed an improved regression estimator using two auxiliary variables with extreme value. The improved regression estimator of ([AK14]) is given as

The corresponding MSE given as

                                                     (15)

where  Similarly,

 are the population regression coefficient between  and   and between  and  .

1. Proposed Mixed Estimator (NEV)

This study has extended the ratio and regression estimators of ([KS13]) into mixed estimation without correction for extreme values. It has also extended the number of auxiliary variables from one to two. The proposed mixed estimator and the reviewed estimators were tested theoretically, empirically and with the use of percentage relative efficiency analysis under High maximum Extreme values and Low minimum Extreme values. The correction factor of ([S72]) is used only were necessary.

The proposed estimator (NEV) is presented as:

The relative error terms are defined as

such that

This implies that

Substituting equation (17) into equation (16), gives

Applying Taylor series, and expanding   up to 2nd order of degree

Substituting equation (19) into equation (20), gives

               since , this implies that

This implies that

Applying expectation,

To obtain the  , differentiate equation (22) and equate to zero.

To obtain the  , substitute equation (23) into equation (22)

                                           Or

2. RESULTS AND DISCUSSIONS

4.1 Theoretical Analysis

The theoretical comparison of the proposed estimators with the reviewed estimators is followed by empirical analysis and percentage relative efficiency analysis. 

The condition for the theoretical analysis is if  .   is more efficient than ; otherwise reverse the decisions in favour of  .

4.11 Comparing MSE of NEV with the MSE of ([KS13]) improved ratio estimator

This implies that  is more efficient than .

4.12 Comparing the MSE of NEV with the MSE of ([AK14]) ratio estimator

This implies that  is more efficient that  

4.13 Comparing the MSE of NEV with the MSE of ([AK14]) regression estimator

This implies that

The efficiency of  over   will be determined empirically using equation (27)

4.2 Empirical Analysis

In the empirical comparison, R statistical software was used to write and compile 728-line code to stimulate and following the normal population of a pre-defined mean and standard deviation of a twenty population. The essence of twenty stimulated population is to test the efficiency of the estimators asymptotically (that is with different populations and sample sizes). Each population has one study variable Y and two auxiliary variables (  ) with the exception of ([KS13]) with one auxiliary variable. The code was developed to compare the estimators under two conditions. The conditions are High Maximum Extreme Value (HMaEV) and Low Minimum Extreme Value (LMiEV).

Table1: Rank and Comparison of the proposed estimator with the reviewed estimators for the twenty stimulated populations for HMaEV cases

Table2: Rank and Comparison of the proposed estimator with the reviewed estimators for the twenty stimulated populations for HMaEV cases continues

Table3: Rank and Comparison of the proposed estimator with the reviewed estimators for the twenty stimulated populations for HMaEV cases continues

 4.3 Low and Minimum Value

Table4: Comparison of the proposed estimators with the reviewed estimators for the twenty stimulated populations for LMiEV

Table5: Comparison of the proposed estimators with the reviewed estimators for the twenty stimulated populations for LMiEV cases (continue)

Table6: Comparison of the proposed estimators with the reviewed estimators for the twenty stimulated populations for LMiEV

4.4 High and Maximum Value

Table 7 :The Relative Efficiency (RE) of estimators developed by ([KS13]) ratio, ([AK14]) regression and ([AK14]) ratio with the proposed estimator for the twenty simulated populations (measured in percentages)