Bayesian inference for the lognormal distribution
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Date
2012
Authors
Harvey, Justin
Journal Title
Journal ISSN
Volume Title
Publisher
University of the Free State
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.
Description
Keywords
Bayesian procedure, Lognormal, MOVER, Credibility intervals, Coverage probabilities, Zero-valued observations, Lognormal variance, Highest posterior density, One-way random effects, Bivariate lognormal, Lognormal distribution, Statistical hypothesis testing, Bayesian statistical decision theory, Derivatives (Mathematics), Observed confidence levels (Statistics), Confidence intervals, Thesis (Ph.D. (Mathematical Statistics and Actuarial Science))--University of the Free State, 2012