# Journal of Advanced Statistics

### Maximum L*q*-likelihood Estimation for Gamma Distributions

Download PDF (1161.4 KB) PP. 54 - 70 Pub. Date: March 1, 2017

### Author(s)

**Jingjing Wu**^{*}

Department of Mathematics and Statistics, University of Calgary, Calgary, Canada**Nana Xing**

Department of Mathematics and Statistics, University of Calgary, Calgary, Canada**Shawn Liu**

Department of Mathematics and Computing, Mount Royal University, Calgary, Canada

### Abstract

*q*E is an extension of MLE which introduces a distortion parameter q to make the estimation more adaptive. The purpose of this work is to examine this methodology for gamma distributions that are widely used in engineering, science and business to model continuous but skewed distributions. For specifically standard gamma models, we look at the ML

*q*E’s asymptotics, finite sample performance in terms of efficiency and robustness, and the choice of the distortion parameter q. We investigate these aspects of ML

*q*E and compare it with MLE in parameter estimation and tail probability estimation, through both Monte Carlo simulation and a real data analysis. Our results show that, with appropriately chosen q, ML

*q*E and MLE perform competitively for large sample sizes while ML

*q*E outperforms MLE for small or moderate sample sizes in terms of reducing MSE. In addition, ML

*q*E with q < 1 has much better robustness properties than MLE when outlying observations are present.

### Keywords

_{q}-likelihood estimation, gamma distribution, distortion parameter, efficiency, robustness.

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