Inference in flexible families of distributions with normal kernel

Abstract

This paper addresses the inference problem for a flexible class of distributions with normal kernel known as skew-bimodal-normal family of distributions. We obtain posterior and predictive distributions assuming different prior specifications. We provide conditions for the existence of the maximum-likelihood estimators (MLE). An EM-type algorithm is built to compute them. As a by product, we obtain important results related to classical and Bayesian inferences for two special subclasses called bimodal-normal and skew-normal (SN) distribution families. We perform a Monte Carlo simulation study to analyse behaviour of the MLE and some Bayesian ones. Considering the frontier data previously studied in the literature, we use the skew-bimodal-normal (SBN) distribution for density estimation. For that data set, we conclude that the SBN model provides as good a fit as the one obtained using the location-scale SN model. Since the former is a more parsimonious model, such a result is shown to be more attractive.

Keywords:

• Bayes estimator
• bimodality
• density estimation
• maximum likelihood
• skewness

Acknowledgements

The authors thank the editor and two referees whose comments and suggestions have contributed to the improvement of the paper. G.H.M.A Rocha acknowledges CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) of the Ministry for Education, Brazil, for partially allowing his research. Rosangela H. Loschi's research has been partially funded by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) of the Ministry for Science and Technology of Brazil, grants 306085/2009-7, 304505/2006-4, 473163/2010-1. The research of R. B. Arellano-Valle was supported in part by FONDECYT (Chile), grant 1085241.