Bayesian inference for the lognormal distribution

Abstract

This thesis is concerned with objective Bayesian analysis (primarily estimation hypothesis testing and confidence statements) of data that are lognormally distributed. The lognormal distribution is currently used extensively to describe the distribution of positive random variables that are right-skewed. This is especially the case with data pertaining to occupational health and other biological data. In Chapter 1 we begin with inference on the products of means and medians as discussed in Menzefricke (1991). Exposure risk modeling is a particular application of this setting. Exact posterior moments are derived and compared to the Monte Carlo simulation techniques. Chapters 2 to 4 are concerned with inference on the mean of the lognormal distributtions in various settings. Other authors, namely Zou, Taleban and Huo (2009), have proposed procedures involving the so-called "method of variance estimates recovery" (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this thesis we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tailed and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (independence Jeffreys' prior, the Jeffreys-rule prior, namely, the square root of the determinant of the Fisher Information matrix, Reference and Probability-Matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidenceintervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In Chapter 5, the variance of the lognormal distribution is the central focus. Similarly to previous chapters, various prior distributions are tested in different applications. In the 6th chapter of this thesis the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors. Chapters 7 and 8 are an investigation into Bayesian methods for analysing the one-way random effects model. Chapter 7 presents the Bayesian framework and results for the balanced model and Chapter 8 is an extension of this setting for the unbalanced model. A new prior distribution, namely Gelman's prior (Gelman, 2006), is introduced.