Journal of Probability and Statistics

Volume 2016, Article ID 3509139, 7 pages

http://dx.doi.org/10.1155/2016/3509139

## A Mixture of Generalized Tukey’s Distributions

Department of Statistics, Universidad Nacional de Colombia, Carrera 45 No. 26-85, Bogotá, Colombia

Received 8 April 2016; Revised 1 July 2016; Accepted 14 July 2016

Academic Editor: Chin-Shang Li

Copyright © 2016 José Alfredo Jiménez and Viswanathan Arunachalam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Mixtures of symmetric distributions, in particular normal mixtures as a tool in statistical modeling, have been widely studied. In recent years, mixtures of asymmetric distributions have emerged as a top contender for analyzing statistical data. Tukey’s family of generalized distributions depend on the parameters, namely, , which controls the skewness. This paper presents the probability density function (pdf) associated with a mixture of Tukey’s family of generalized distributions. The mixture of this class of skewed distributions is a generalization of Tukey’s family of distributions. In this paper, we calculate a closed form expression for the density and distribution of the mixture of two Tukey’s families of generalized distributions, which allows us to easily compute probabilities, moments, and related measures. This class of distributions contains the mixture of Log-symmetric distributions as a special case.

#### 1. Introduction

The main focus of interest in financial economics is the distribution of stock market returns. Mandelbrot [1] suggested the family of stable Paretian distributions for stock market returns. Fama [2] established that the normality assumption of the empirical data does not hold as the distribution is fat tailed. Kon [3] and Tse [4] used a mixture of normal distributions for stock return. Fielitz and Rozelle [5] proposed a mixture of nonnormal stable distributions for stock price. Consequently, greater emphasis has been placed on using distributions which have asymmetry and leptokurtic properties. Recently Jiménez et al. [6] proposed option pricing based mixture of log-skew-normal distributions. If extreme events tend to occur more frequently than normal events, then skewness and kurtosis of nonnormal distributions play an essential role for the volatility smile.

The most important and useful characteristic of Tukey’s family of distributions introduced by Tukey [7] is that it covers most of the Pearsonian family of distributions. It can also generate several known distributions, for example, lognormal, Cauchy, exponential, and Chi-squared (see Martínez and Iglewicz [8], page 363). From Tukey’s family of distribution, we obtain distribution, which is closely related to lognormal distribution and possesses similar properties of moments. Tukey’s family of distributions have been used to study financial markets. Badrinath and Chatterjee [9, 10] and Mills [11] used to model the return on a stock index, as well as the return on shares in several markets. Dutta and Babbel [12] found that the skewness and leptokurtic behavior of LIBOR were modeled effectively using distribution. Dutta and Babbel [13] used to model interest rates and options on interest rates, while Tang and Wu [14] proposed a new method for the Decomposition of Portfolio VaR. Dutta and Perry [15] and recently Jiménez and Arunachalam [16] used distribution to study the operational risk for heavy tailed severity models. Jiménez and Arunachalam [17] provided explicit expressions for VaR and CVaR calculations using the family of Tukey’s distributions. Currently Jiménez et al. [18] studied generalization of Tukey’s family of distributions, when the standard normal random variable is replaced by a continuous random variable with mean and variance

The subfamily of distributions exhibits skewness and has great importance in the study of asymmetric distributions for analyzing data. This kind of distribution allows us to obtain scaled Log-symmetric distributions. Vitiello and Poon [19] considered a simple mixture of two distributions for option pricing data. The purpose of this paper is to present a mixture of Tukey’s distributions and derive some statistical properties including the pdf and moment generating function and its properties.

The paper is organized as follows: Section 2 presents Tukey’s family of generalized distributions and its pdf, as well as the cumulative distribution function (cdf). In Section 3, some theoretical results of the mixture of two Tukey’s families of generalized distributions are presented and Section 4 explains the methodology of calculating estimation of parameters by the method of moments. Section 5 discusses the adjustment methodology of our proposed model to real data of Heating-Degree-Days (HDD) indices and finally, in Section 6, we conclude.

#### 2. Tukey’s Family of Generalized Distributions

Tukey [7] introduced the family distributions by means of two nonlinear transformations given bywith , where the distribution of is standard normal. When these transformations are applied to a continuous random variable with mean and variance such that its pdf is symmetric about the origin and cdf , the transformation is obtained, which henceforth will be termed Tukey’s generalized distribution. If , Tukey’s generalized distribution reduces towhich is known as Tukey’s generalized distribution.

In order to model an arbitrary random variable using the transformation given in (2), Hoaglin and Peters [20] introduced two new parameters, (location) and (scale), and proposed the following linear transformation:The following properties for pdf, cdf, and quantile functions of Tukey’s generalized distribution were established by Jiménez et al. [18] in terms of the and of as follows:where and . We say that the random variable has a Log-symmetric distribution (such distributions are all asymmetric; see for reference Johnson et al. [21] and Stuart and Ord [22]) with three parameters: threshold , scale , and shape , denoted by

The first expression of (4) allows us to obtain the following pdf associated with Tukey’s distribution. Table 1 shows the parameters of the pdf of that we obtain using a selected set of well known symmetrical distributions (from Jiménez et al. [18]).