REJOINDER: EM-BASED LIKELIHOOD INFERENCE FOR SOME LIFETIME DISTRIBUTIONS BASED ON LEFT TRUNCATED AND RIGHT CENSORED DATA AND ASSOCIATED MODEL DISCRIMINATION

摘要

First of all, we express our sincere thanks to Drs. Laurent Bordes of Universite de Pau et des Pays de l'Adour and Didier Chauveau of Universite d'Orleans, Dr. Isha Dewan of Indian Statistical Institute at New Delhi, Drs. Hon Keung Tony Ng of Southern Methodist University and Zhisheng Ye of Hong Kong Polytechnic University, Drs. Yili Hong and Caleb King of Virginia Tech, Drs. Iain L. MacDonald and Brendon M. Lapham of University of Cape Town, Dr. Tertius de Wet of Stellen-bosch University, and Dr. Hideki Nagatsuka of Chuo University for writing insightful discussions on our invited paper. Their valuable discussions certainly further the topic of discussion of our paper by providing some additional insight into the topic and also by adding some more directions of future research in the analysis of left truncated and right censored data. Drs. Ng and Ye and Drs. Bordes and Chauveau have discussed the stochastic-EM algorithm in the context considered in our paper. While Drs. Bordes and Chauveau have discussed the stochastic-EM algorithm for the case of Weibull lifetime distribution, Drs. Ng and Ye have developed the stochastic-EM algorithm for the generalized gamma distribution, both under left truncated and right censored data. In both these discussions, the stochastic-EM algorithm has been explained clearly, and the specific steps for Weibull and generalized gamma distributions have been developed in a careful and comprehensive manner. For the Weibull distribution, it is seen that the results obtained by Drs. Bordes and Chauveau are quite close to those obtained by us. However, for the generalized gamma distribution with left truncation and right censoring, Drs. Ng and Ye have pointed out that the EM algorithm may converge to a local maxima. In this case, the stochastic-EM algorithm clearly provides a better alternative, as it avoids getting trapped into any saddle point. Our special thanks go to Drs. Ng and Ye for pointing out this issue with the EM algorithm for left truncated and right censored data from the generalized gamma distribution.