Summary
In this paper we investigate a Bayesian procedure for the estimation of a flexible generalised distribution, notably the MacGillivray adaptation of theg-and-k distribution. This distribution, described through its inverse cdf or quantile function, generalises the standard normal through extra parameters which together describe skewness and kurtosis. The standard quantile-based methods for estimating the parameters of generalised distributions are often arbitrary and do not rely on computation of the likelihood. MCMC, however, provides a simulation-based alternative for obtaining the maximum likelihood estimates of parameters of these distributions or for deriving posterior estimates of the parameters through a Bayesian framework. In this paper we adopt the latter approach. The proposed methodology is illustrated through an application in which the parameter of interest is slightly skewed.
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Acknowledgements
We thank Dr J. Wood of the Centre for Eye Research, QUT, for the data on visual performance used in the example, and Associate Professor H. L. MacGillivray of the School of Mathematical Sciences, QUT, for useful discussions and suggestions. We also thank Dr G. Rayner, a Quantitative Analyst with the National Australia Bank, Melbourne, Australia, for assistance with generating the data samples and producing Figure 1. Comments and suggestions from anonymous referees, which contributed to a much stronger revised version of this paper, are gratefully acknowleged.
Most of the computations referred to in this Chapter were carried out on computing equipment supplied to the Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, under the Digital Equipment Agreement ERP No 2057.
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Haynes, M., Mengersen, K. Bayesian estimation ofg-and-k distributions using MCMC. Computational Statistics 20, 7–30 (2005). https://doi.org/10.1007/BF02736120
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DOI: https://doi.org/10.1007/BF02736120