Improved Mixed Estimator Using Two Auxiliary Variables For Full Extreme Maximum And Minimum Values In Single Phase Sampling

     This study has proposed improved mixed estimator in the presence of extreme values using two auxiliary variables in single-phase sampling. The theoretical analysis shows that comparing the proposed estimator FEVSA (  with the ratio estimator of Khan and Shabbir (2013), (   and  the ratio estimator of Al-Hossain and Khan (2014), ( , using their mean square errors, equations 28 and 29 both makes it obvious that FEVSA (  is superior to these estimators. Lastly, comparing FEVSA (  with the regression estimator of Al-Hossain and Khan (2014), (  using equation 30, could not be concluded; but later resolved using empirical analysis.

    In the empirical analysis, the R statistical programming language was used to performed exploratory Data Analysis (EDA) for each of the twenty- populations and to summarize the main characteristics (not with visual statistical tools) of the distributions. The function code asymptotically computed the bias, Mean Square Error (MSE) and variance for the proposed and reviewed estimators in each of the twenty stimulated population.                                    

Table 1 through 3 show that the Mean Square Error (MSE) obtained for the proposed estimator is smaller compared to that of the revealed estimators for the cases of HMaEV; this implies that the proposed estimator is more efficient than the reviewed. Likewise, table 4 through 6, agreed with table 1 to 3, that the proposed estimator is superior to the reviewed estimators. Using the ranks for the HMaEV cases, table 3 (the overall rank table for the HMaEV) shows that the proposed estimator FEVSA ( , is ranked first and hence more effective than the reviewed estimators. This is also supported by table 6 (the overall rank table for the LMiEV cases). The empirical analysis revealed that the proposed FEVSA ( , outperformed all the considered estimators for HMaEV and LMiEV cases.

    Finally, the function code computed the Relative Efficiency (RE) of the proposed estimator to the reviewed estimators. This answer the question that says by what percentage is the proposed estimator efficient over the reviewed estimators. The percentage relative efficiency Table 9, reveals that the proposed estimator FEVSA,  is 6.3803%, 4375.686%, 66.2312% and 39.7252% relatively efficient over ,  and  respectively for the HMaEV cases. Likewise, table 12 reveals that  is 12.361%, 3553.709%, 74.9976% and 0.6684% relatively efficient over ,  and  respectively for the LMiEV cases.  This implies that the proposed estimator   is asymptotically more efficient over all the estimators considered in this study irrespective of the type of extreme value case. Therefore, the proposed estimator is recommended subject to the validation of the condition of usage.