The Kummer Beta Normal: A New Useful-Skew Model

Rodrigo R Pescim,Saralees Nadarajah

doi:10.6339/jds.201507_13(3).0006

Abstract

The normal distribution is the most popular model in applications to real data. We propose a new extension of this distribution, called the Kummer beta normal distribution, which presents greater flexibility to model scenarios involving skewed data. The new probability density function can be represented as a linear combination of exponentiated normal pdfs. We also propose analytical expressions for some mathematical quantities: Ordinary and incomplete moments, mean deviations and order statistics. The estimation of parameters is approached by the method of maximum likelihood and Bayesian analysis. Likelihood ratio statistics and formal goodnessof-fit tests are used to compare the proposed distribution with some of its sub-models and non-nested models. A real data set is used to illustrate the importance of the proposed model.

Highlights

The main motivation for statisticians to study new families of statistical distributions is to increase the flexibility to better model various data sets that cannot be properly fitted by the existing distributions
Several authors have proposed new generalizations based on the normal distribution for modeling real data sets; see, for example, Azzalini (1985), Eugene et al (2002), Nadarajah (2005), Cordeiro and de Castro (2011), Cordeiro et al (2012a), Nadarajah et al (2014) and Alzaatreh et al (2014) for skew-normal, beta normal, generalized normal, Kumaraswamy normal, McDonald normal, modified beta normal and gamma normal distributions, respectively
We have introduced the Kummer beta normal (KBN) distribution with three shape parameters

Summary

Introduction

The main motivation for statisticians to study new families of statistical distributions is to increase the flexibility to better model various data sets that cannot be properly fitted by the existing distributions. Several authors have proposed new generalizations based on the normal distribution for modeling real data sets; see, for example, Azzalini (1985), Eugene et al (2002), Nadarajah (2005), Cordeiro and de Castro (2011), Cordeiro et al (2012a), Nadarajah et al (2014) and Alzaatreh et al (2014) for skew-normal, beta normal, generalized normal, Kumaraswamy normal, McDonald normal, modified beta normal and gamma normal distributions, respectively The emergence of such distributions in the statistics literature is only very recent.

Useful expansions

Order Statisitcs

Bayesian inference

Application – INPC data

Concluding remarks