Isaac Scientific Publishing

Journal of Advanced Statistics

Maximum Lq-likelihood Estimation for Gamma Distributions

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

DOI: 10.22606/jas.2017.21007


  • 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


In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model. Standard large sample theory guarantees asymptotic efficiency of MLE. On the other hand, MLE does not perform as well as expected for moderate or small sample size. In 2010, a new parameter estimator based on nonextensive entropy ([1]), named Maximum Lq-likelihood Estimator (MLqE), was first introduced and studied by [2]. MLqE 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 MLqE’s asymptotics, finite sample performance in terms of efficiency and robustness, and the choice of the distortion parameter q. We investigate these aspects of MLqE 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, MLqE and MLE perform competitively for large sample sizes while MLqE outperforms MLE for small or moderate sample sizes in terms of reducing MSE. In addition, MLqE with q < 1 has much better robustness properties than MLE when outlying observations are present.


Maximum Lq-likelihood estimation, gamma distribution, distortion parameter, efficiency, robustness.


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