Weights Sum of Squares Ratio as an Alternative to Relative Efficiency in Selecting the Best Linear Unbiased Estimator of Population Mean for Correlated Random Variables
E. Onyemachi *
Federal University of Technology, Owerri, Imo State, Nigeria.
E. W. Okereke
Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
I. S. Iwueze
Federal University of Technology, Owerri, Imo State, Nigeria.
C. O. Omekara
Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
Weights Sum of Squares Ratio (WSSR) is proposed in this paper to be used as an alternative to relative efficiency in selecting the best linear unbiased estimator (BLUE) of the population mean for correlated random variables with known covariance structure. In the realm of statistical computing, the weights assigned to the linear unbiased estimator (LUE) of population mean significantly influence its stability. The weights guarantee that LUE are unbiased and have minimal or reduced variance across comparable classes of estimators. For correlated random variables with a moving average process of order one structure, the Langrangian Multiplier Symmetric Based Modified (LMSBM) approach was used to calculate the efficient weights of the generalized linear unbiased estimator (GLUE) of the population mean. The sufficient condition, which provided theoretical insights, for algebraic equivalency between two LUEs (GLUE and arithmetic mean (AM)) of the population mean was found using the geometric properties of the vector of weights. These insights contributed to the creation of a novel statistical method that assesses the efficiency of GLUE relative to AM by means of WSSR, as opposed to variance-based relative efficiency. The findings suggest that when using WSSR or variance-based relative efficiency for the evaluation of efficiency, the correlation point's interval across various sample sizes for GLUE's equi-efficiency with AM converges to approximately [ -0.20, 0.20]. This convergent interval for GLUE's equi-efficiency with AM matches the interval, which was calculated using WSSR for judgment. WSSR is favored as an alternative for confirming GLUE's efficient performance in comparison to AM because it is easy to calculate and yields the same conclusion as variance-based relative efficiency. Through real-life practical application, the usefulness of WSSR as a basis for evaluating efficiency was shown.
Keywords: Arithmetic Mean (AM), Linear Unbiased Estimator (LUE), Generalized Linear Unbiased Estimator (GLUE), Relative Efficiency, Weights Sum of Squares Ratio (WSSR)