Inadmissibility of invariant estimator of function of scale parameter of several exponential distributions

TL;DR

This paper investigates the estimation of the entropy of multiple exponential distributions with unknown location parameters and a common scale, demonstrating the inadmissibility of the invariant estimator and proposing improved alternatives.

Contribution

It introduces non-smooth and smooth improved estimators for entropy, proving inadmissibility of the invariant estimator and extending results to censored sampling schemes.

Findings

Proved the inadmissibility of the minimum risk invariant estimator.

Developed improved estimators under squared error and linex loss functions.

Extended estimators to censored sampling schemes such as record values and Type-II censoring.

Abstract

In various applied areas such as reliability engineering, molecular biology, finance, etc., the measure of uncertainty of a probability distribution plays an important role. In the present work, we consider the estimation of a function of the scale parameter, namely entropy of many exponential distributions having unknown and unequal location parameters with a common scale parameter. For this estimation problem, we have considered bowl-shaped location invariant loss functions. The inadmissibility of the minimum risk invariant estimator (MRIE) is proved by proposing a non-smooth improved estimator. Also, we have obtained a smooth estimator which improves upon the MRIE. As an application, we have obtained explicit expressions of improved estimators for two well-known loss functions namely squared error loss and linex loss. Further, we have shown that these estimators can be derived for other important censored sampling schemes. At first, we obtained the results for the complete and i.i.d. sample. We have seen that the results can be applied for (i) record values, (ii) type-II censoring, and (iii) progressive Type-II censoring. Finally, a simulation study has been carried out to compare the risk performance of the proposed improved estimators.