Inference Using Latent Variables for Mixtures of Distributions for Censored Data with Partial Identification

Abstract

In this article two methods are proposed to make inferences about the parameters of a finite mixture of distributions in the context of partially identifiable censored data. The first method focuses on a mixture of location and scale models and relies on an asymptotic approximation to a suitably constructed augmented likelihood; the second method provides a full Bayesian analysis of the mixture based on a Gibbs sampler. Both methods make explicit use of latent variables and provide computationally efficient procedures compared to other methods which deal directly with the likelihood of the mixture. This may be crucial if the number of components in the mixture is not small. Our proposals are illustrated on a classical example on failure times for communication devices first studied by Mendenhall and Hader (Mendenhall, W., Hader, R. J. (1958). Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45:504–520.). In addition, we study the coverage of the confidence intervals obtained from each of the methods by means of a small simulation exercise.

Keywords:

• Bayesian inference
• EM algorithm
• Extreme value distribution
• Fiducial inference
• Gibbs sampler
• Profile likelihood
• Reliability and life testing

Acknowledgments

We are grateful to Dongchu Sun for his remarks concerning the propriety of the posterior density (6). We would also like to thank an anonymous referee and an Associate Editor for their thorough reviews which greatly improved the presentation of the article. This work was supported by CONACyT Grants 32256-E and 32705-E. Partial support from the Sistema Nacional de Investigadores is gratefully acknowledged.