On improved estimation of a gamma shape parameter

Abstract

This paper deals with improved estimation of a gamma shape parameter from a decision-theoretic point of view. First we study the second-order properties of three estimators – (i) the maximum-likelihood estimator (MLE), (ii) a bias corrected version of the MLE, and (iii) an improved version (in terms of mean squared error) of the MLE. It is shown that all the three estimators mentioned above are second-order inadmissible. Next, we obtain superior estimators which are second order better than the above three estimators. Simulation results are provided to study the relative risk improvement of each improved estimator over the MLE.

Keywords:

• gamma shape parameter
• second-order admissibility (inadmissibility)
• modified maximum-likelihood estimator

AMS 2000 Subject Classifications:

• 65F12
• 62C15
• 62F03

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CORRIGENDUM

Acknowledgements

We would like to thank the two anonymous referees who went over the first draft of this work meticulously and made constructive comments. This has helped us in improving the presentation of this work. The authors would also like to acknowledge the paper by Takagi Y. On the estimation of the shape parameter of the gamma distribution in second-order asymptotics. Statistics & Probability Letters 1981;9:1334-1338 and the paper by Ghosh JK, Sinha BK. A necessary and sufficient condition for second order admissibility with applications to Berkson's bioassay problem. Ann Statist. 1981; 9:1334–1338.

Funding

The first author's research was partially supported by Grant-in-Aid for Young Scientists (B) [grant number 21700309].