A Bayesian approach to inference on the variance of lognormal data

Abstract

Krishnamoorthy, Mathewand Ramachandran (2006) developed a method to draw inference on the mean and variance of one or more lognormal distributions. Their method was based on generalised confidence intervals (frequentist methods). In this article we focus on the variance of the lognormal distribution and implement a Bayesian approach and obtain credibility intervals to compare the performance of four different non-informative prior distributions. This is done by means of various Monte Carlo simulation studies as well as practical examples. The accuracy (coverage) and efficiency (interval length) of some of the Bayesian priors, particularly for the highest posterior density (HPD) credibility intervals will be illustrated in these simulation studies and examples. It can be observed that the frequentist approach is equivalent to the Bayesian approach, when using the Independence Jeffreys prior. Even so, the Bayesian approach offers some additional benefits, namely, through the calculation of the HPD intervals. Hypothesis testing and practical applications are also presented. Further results comparing various estimators of the lognormal variance are derived and evaluated. The usefulness of the Bayesian approach is also illustrated in its ability to easily modify the method to account for the possibility of zero-valued observations. This is something for which there is (to our knowledge) currently no frequentist method available and serves to highlight the usefulness of the Bayesian approach.