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

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