Aspects of inference in the Birnbaum-Saunders and Sinh-Normal distributions

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Cíntora González, Carlos L.

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University of Guelph

Abstract

In this dissertation we consider methods of inference for the Sinh-Normal and Birnbaum-Saunders distributions. The methods of inference presented include the cases of complete observations and Type-I censored observations. Estimation based on the likelihood function for complete observations is presented and discussed first. Then, the behavior of maximum likelihood estimators and confidence intervals based on the Wald, likelihood ratio statistic, parametric bootstrap, and two higher accuracy likelihood based methods for constructing confidence intervals are studied via Monte Carlo simulation; the discussion assumes Type-I censoring. A stochastic version of the EM algorithm is applied to obtain the maximum likelihood estimates under Type-I censoring. A log-linear Birnbaum-Saunders mixed model is constructed and methods of inference for it are presented as well. Monte Carlo simulations to study the behavior of estimates of the fixed effects are implemented and their results discussed. The conservativeness of the Wald test, using the proposed reference distribution, is studied, as are the coverage and bias of the confidence intervals based on such a model. An example of application to a real data set is presented. In part two, Bayesian methods of inference are presented for the case of complete and Type-I censored observations. Independent non-informative priors, based on the Jeffreys criterion, are obtained and used to derive highest posterior density regions (HPDR). Methods for approximating and studying the marginal posteriors are presented and discussed, and Bayesian analyses of real data sets are presented to show the application of such methods. An application of the data augmentation algorithm to a real data set, under Type-I censoring, is done; a log-linear Birnbaum-Saunders regression model and non-informative priors are assumed. Results from a Monte Carlo simulation experiment designed to study the coverage of the HPDR, obtained from the independent non-informative priors, are presented. In the last part, some methods for the analysis of residuals and to assess the goodness of fit are presented.

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Keywords

Sinh-Normal distribution, Birnbaum-Saunders distribution, inference, complete observations, Type-I censored observations, likelihood function

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