Elsevier

Statistics & Probability Letters

Volume 104, September 2015, Pages 75-81
Statistics & Probability Letters

The Kumaraswamy skew-normal distribution

https://doi.org/10.1016/j.spl.2015.04.031Get rights and content

Abstract

We propose a new generalization of the skew-normal distribution (Azzalini, 1985) referred to as the Kumaraswamy skew-normal. The new distribution is computationally more tractable than the Beta skew-normal distribution (Mameli and Musio, 2013) with which it shares some properties.

Introduction

Some recent developments in distribution theory have proposed new techniques for building distributions. Among these, the methods used to construct the Beta generalized (BetaF) (Jones, 2004) and the Kumaraswamy generalized (KwF) (Cordeiro and de Castro, 2011) class of distributions have received a lot of attention. The first work concerning the Beta-generated family was proposed by Eugene et al. (2002), who defined and analysed the Beta-normal distribution. Further, Jones (2004) formalized the definition of the Beta-generated family. Its work has inspired many researchers and has fuelled an enormous literature regarding this family of distributions; see for example Gupta and Nadarajah (2005), Pescim et al. (2010) and Mameli and Musio (2013). Recently, following the idea of the class of Beta-generated distributions (Jones, 2004), Cordeiro and de Castro (2011) proposed a new family of generalized distributions, called Kumaraswamy generalized family, by means of the Kumaraswamy distribution (Kumaraswamy, 1980, Jones, 2009). The maximum likelihood estimation for the family KwF distribution results simpler than the estimation in the BetaF family. Motivated by these facts, we define in this paper a new generalization of the skew-normal based on the Kumaraswamy generalized family, which is more tractable of the Beta skew-normal (BSN) introduced by Mameli and Musio (2013). The resulting distribution, which will be called the Kumaraswamy skew-normal (KwSN), could be considered a valid alternative to the BSN distribution with which it shares some similar properties. Moreover, for special values of the parameters the KwSN distribution is related to the Beta skew-normal one. The KwSN distribution is always unimodal, unlike the Beta skew-normal which can be either unimodal or bimodal. The KwSN model shows more flexibility than the SN one. Furthermore, under the null hypothesis of normality the KwSN distribution, as the BSN one, is not identifiable. However, due to the tractability of the KwSN density, all the possible sets of parameters for which this density reduces to the normal one can be established by exploiting the Lambert W function; see Jeffrey et al. (1998). The rest of the paper organizes as follows. Section  2 defines the KwSN distribution and presents some properties of the new distribution. Section  3 investigates maximum likelihood estimation and analyses a data set of Australian athletes measurements. Finally, concluding remarks are given in Section  4.

Section snippets

The new model

In this section we first define the Kumaraswamy skew-normal distribution and then we present some of its properties.

Estimation

As we have seen in the previous section, the parameters representing the true null distribution are not unique in the Gaussian case and classical likelihood result does not apply (see e.g. Liu and Shao, 2003 and references therein). Hence, we just consider a special subclass of this family by choosing b=1a, this choice leads to a model which under the null hypothesis of normality is described only by the parameters a=1,b=1 and λ=0.

Consider a sample x1,,xN from the KwSN(a,1a,μ,σ,λ) density. The

Final remarks

In this paper, we study some structural properties of a new generalization of the skew-normal distribution, called the Kumaraswamy skew-normal distribution, which represents a valid alternative to the Beta skew-normal one (Mameli and Musio, 2013). Both distributions have the normal and skew-normal distributions as special cases. The Kumaraswamy skew-normal model, as well as the Beta skew-normal one, presents problems of identifiability under the null hypothesis of normality. Due to the

Acknowledgements

The author thanks Monica Musio for reading previous drafts of the work and Stefano Montaldo for the help in implementing with Mathematica the system of Eqs. (9) in the supplementary material (see Appendix A). This research was partially supported by a grant from the University of Cagliari grant no. F71J12001120002. The author thanks the editor and an anonymous referee for very helpful comments and suggestions. This work was carried out while the author was at Department of Mathematics and

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