Statistics and Probability Letters 81 (2011) 1813–1821 Contents lists available at SciVerse ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro On a generalized mixture of standard normal and skew normal distributions C. Satheesh Kumar ∗ , M.R. Anusree Department of Statistics, University of Kerala, Trivandrum-695 581, India article info Article history: Received 17 February 2011 Received in revised form 6 July 2011 Accepted 6 July 2011 Available online 14 July 2011 Keywords: Characteristic function Method of maximum likelihood Skew normal distribution Plurimodality abstract Here we propose a new class of distributions as a generalized mixture of standard normal and skew normal distributions (GMNSND) and study some of its properties by deriving its characteristic function, mean, variance, coefficient of skewness etc. Further, certain reliability aspects of GMNSND are studied and a location scale extension of GMNSND is considered. The estimation of the parameters of this extended GMNSND by the method of maximum likelihood is discussed. © 2011 Elsevier B.V. All rights reserved. 1. Introduction The normal distribution is the basis of many statistical work and has a unique position in probability theory. It is an unavoidable tool for the analysis and interpretation of basic data. It can be noted that the unrestricted usage of normal distribution to model a data in many real life applications leads to an error in the result. This may be due to the effect of certain unknown variables giving rise to skewness in the data. To overcome this difficulty, Azzalini (1985) introduced a generalized version of normal distribution, namely skew normal distribution and this distribution has been studied by several authors such as Azzalini (1986), Henze (1986), Azzalini and Dalla-Valle (1996) and Branco and Dey (2001). Azzalini (1985) defined skew normal distribution as in the following. A random variable Z is said to have skew normal distribution with parameter λ ∈ R = (−∞, ∞) if its probability density function (p.d.f.) g (z ; λ) is of the following form. For z ∈ R, g (z ; λ) = 2f (z )F (λz ) (1.1) where f (.) and F (.) are, respectively, the p.d.f. and cumulative distribution function (c.d.f.) of a standard normal variate. Hereafter, we denote a distribution with p.d.f. (1.1) as SND(λ). The density given by (1.1) is appropriate for the data exhibiting unimodal density with some skewness present in it. Also it has certain formal properties which hold for normal distribution. Buccianti (2005) remarked that normal and skew normal models are not adequate to describe the situations of plurimodality. He investigated the shape of the frequency distribution of the logratio ln(cl − /Na + ) whose components are related to water composition for 26 wells. Samples have been collected around the active center of Vulcano Island from 1977. Data of the logratio have been tentatively modeled by evaluating the performance of the skew normal model for each well. Value of λ for the wells of Vulcano Island appear to cover a wide range corresponding to (1) a more or less good symmetry, (2) the presence of a moderate skewness, (3) the presence of plurimodality. For the first and second situation, he noted ∗ Corresponding author. Tel.: +91 04712418905. E-mail addresses: drcsatheeshkumar@gmail.com (C. Satheesh Kumar), anusreemr@yahoo.co.in (M.R. Anusree). 0167-7152/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2011.07.009