Journal of Advanced Statistics

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 L_q-likelihood estimation, gamma distribution, distortion parameter, efficiency, robustness.

[1] J. Havrda and F. Charvát, “Quantification method of classification processes. concept of structural a-entropy,” Kybernetika, vol. 3, no. 1, pp. 30–35, 1967.

[2] D. Ferrari and Y. Yang, “Maximum lq-likelihood estimation,” The Annals of Statistics, vol. 38, no. 2, pp. 753–783, 2010.

[3] H. Akaike, “Information theory and an extension of the maximum likelihood principle,” in Selected Papers of Hirotugu Akaike. Springer, 1998, pp. 199–213.

[4] A. Barron, J. Rissanen, and B. Yu, “The minimum description length principle in coding and modeling,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2743–2760, 1998.

[5] J. Aczél and Z. Daróczy, On measures of information and their characterizations. AP, 1975.

[6] A. Rényi et al., “On measures of entropy and information,” in Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, vol. 1, 1961, pp. 547–561.

[7] E. T. Jaynes, “Information theory and statistical mechanics,” Physical review, vol. 106, no. 4, pp. 620–630, 1957.

[8] E. T. Jaynes, “Information theory and statistical mechanics. ii,” Physical review, vol. 108, no. 2, pp. 171–190, 1957.

[9] C. Tsallis, “Possible generalization of boltzmann-gibbs statistics,” Journal of statistical physics, vol. 52, no. 1, pp. 479–487, 1988.

[10] C. Tsallis, R. Mendes, and A. R. Plastino, “The role of constraints within generalized nonextensive statistics,” Physica A: Statistical Mechanics and its Applications, vol. 261, no. 3, pp. 534–554, 1998.

[11] Y. Altun and A. Smola, “Unifying divergence minimization and statistical inference via convex duality,” in International Conference on Computational Learning Theory. Springer, 2006, pp. 139–153.

[12] Y. Qin and C. E. Priebe, “Maximum lq-likelihood estimation via the expectation-maximization algorithm: A robust estimation of mixture models,” Journal of the American Statistical Association, vol. 108, no. 503, pp. 914–928, 2013.

[13] D. Ferrari and S. Paterlini, “The maximum lq-likelihood method: an application to extreme quantile estimation in finance,” Methodology and Computing in Applied Probability, vol. 11, no. 1, pp. 3–19, 2009.

[14] C. Huang, J.-G. Lin, and Y.-Y. Ren, “Testing for the shape parameter of generalized extreme value distribution based on the lq-likelihood ratio statistic,” Metrika, vol. 76, no. 5, pp. 641–671, 2013.

[15] R. Beran, “Minimum hellinger distance estimates for parametric models,” The Annals of Statistics, pp. 445–463, 1977.

[16] Z. Lu, Y. V. Hui, and A. H. Lee, “Minimum hellinger distance estimation for finite mixtures of poisson regression models and its applications,” Biometrics, vol. 59, no. 4, pp. 1016–1026, 2003.

[17] T. Bjerkedal, “Acquisition of resistance in guinea pies infected with different doses of virulent tubercle bacilli.” American Journal of Hygiene, vol. 72, no. 1, pp. 130–148, 1960.