Departamento de Estadística Universidad Carlos III de Madrid Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624-98-49

BAYESIAN INFERENCE FOR THE HALF-NORMAL AND HALF-T DISTRIBUTIONS M.P. Wiper, F.J. Girón, A. Pewsey*

Abstract In this article we consider approaches to Bayesian inference for the half-normal and half-t distributions. We show that a generalized version of the normal- gamma distribution is conjugate to the half-normal likelihood and give the moments of this new distribution. The bias and coverage of the Bayesian posterior mean estimator of the halfnormal location parameter are compared with those of maximum likelihood based estimators. Inference for the half-t distribution is performed using Gibbs sampling and model comparison is carried out using Bayes factors. A real data example is presented which demonstrates the fitting of the half-normal and half-t models.

Keywords: Bias-correction; Gaussian-modulated gamma distribution; Gibbs sampling; likelihood based inference; model selection; right-truncated normal- gamma distribution.

* Wiper, Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid ), e-mail: mwiper@est-econ.uc3m.es; Girón, Departamento de Estadística e Investigación Operativa, Universidad de Malaga; Pewsey, Departamento de Matemáticas, Universidad de Extremadura Departamento de Matemáticas, Universidad de Extremadura.

Bayesian inference for the half-normal and half-t distributions M.P. Wiper∗ F.J. Gir´n† & A. Pewsey‡ , o , July 21, 2005

Abstract In this article we consider approaches to Bayesian inference for the halfnormal and half-t distributions. We show that a generalized version of the normal-gamma distribution is conjugate to the half-normal likelihood and give the moments of this new distribution. The bias and coverage of the Bayesian posterior mean estimator of the half-normal location parameter are compared with those of maximum likelihood based estimators. Inference for the half-t distribution is performed using Gibbs sampling and model comparison is carried out using Bayes factors. A real data example is presented which demonstrates the ﬁtting of the half-normal and half-t models. KEY WORDS: Bias-correction; Gaussian-modulated gamma distribution; Gibbs sampling; likelihood based inference; model selection; right-truncated normal-gamma distribution.

1

Introduction

The half-normal distribution has been used as a model for (left) truncated data from application areas as diverse as ﬁbre buckling (Haberle 1991), blowﬂy dispersion (Dobzhansky and Wright 1947), sports science physiology (Pewsey 2002, 2004) and, in particular, stochastic frontier modelling (Aigner et al., 1977; Meeusen and van den Broeck, 1977). Likelihood based inference for the halfnormal distribution has been considered by Pewsey (2002, 2004). However, for heavy-tailed data, the half-normal distribution will not be an adequate model and then a half-t distribution might be considered as a more ﬂexible alternative. For one of the few applications of this latter model, see Tancredi (2002). The principal objective of this article is to illustrate that fully conjugate Bayesian inference can be carried out for the half-normal model and that Gibbs ∗ Departamento † Departamento

de Estad´ ıstica, Universidad Carlos III de Madrid de Estad´ ıstica e Investigaci´n Operativa, Universidad de Malaga o ‡ Departamento de Matem´ticas, Universidad de Extremadura a

1

sampling techniques can be used to perform Bayesian inference for the parameters of the half-t model. The article is structured as follows. In Section 2, we review the deﬁnition of the half-normal distribution and comment on likelihood based inference for its parameters. In Section 3, we illustrate how conjugate Bayesian inference for the half-normal distribution can be undertaken. Results for the posterior moments of the location...