Shrinkage Estimation for Location and Scale Parameters of Logistic Distribution Under Record Values

Abstract

Logistic distribution (LogDis) is frequently used in many different applications, such as logistic regression, logit models, classification, neural networks, physical sciences, sports modeling, finance and health and disease studies. For instance, the distribution function of the LogDis has the same functional form as the derivative of the Fermi function that can be used to set the relative weight of various electron energies in their contributions to electron transport. The LogDis has wider tails than a normal distribution (NorDis), so it is more consistent with the underlying data and provides better insight into the likelihood of extreme events. For this reason the United States Chess Federation has switched its formula for calculating chess ratings from the NorDis to the LogDis. The outcomes of many real-life experiments are sequences of record-breaking data sets, where only observations that exceed (or only those that fall below) the current extreme value are recorded. The practice demonstrated that the widely used estimators of the scale and location parameters of logistic record values, such as the best linear unbiased estimators (BLUEs), have some defects. This paper investigates the shrinkage estimators of the location and scale parameters for logistic record values using prior information about their BLUEs. Theoretical and computational justifications for the accuracy and precision of the proposed shrinkage estimators are investigated via their bias and mean square error (MSE), which provide sufficient conditions for improving the proposed shrinkage estimators to get unbiased estimators with minimum MSE. The performance of the proposed shrinkage estimators is compared with the performances of the BLUEs. The results demonstrate that the resulting shrinkage estimators are shown to be remarkably efficient.