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On Some Properties of a General Class of Two-Piece Skew Normal Distribution

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On Some Properties of a General Class of Two-Piece Skew Normal Distribution J. Japan Statist. Soc. Vol. 44 No. 2 2014 179–194 ON SOME PROPERTIES OF A GENERAL CLASS OF TWO-PIECE SKEW NORMAL DISTRIBUTION C. Satheesh Kumar* and M. R. Anusree* A new class of generalized two-piece skew normal distribution is introduced here as a two-piece version of the generalized skew normal distribution of Kumar and Anusree (2011).It is shown that the proposed class of distribution will be more suitable for modelling skewed, multimodal data sets.Several properties of the model are studied and the maximum likelihood estimation of the parameters of the distribution is discussed.Further, the practical usefulness of the model is illustrated with the help of certain real life data sets. Key words and phrases: Asymmetric distributions, method of maximum likelihood, plurimodality, reliability measures, skew normal distribution. 1. Introduction An important drawback of the normal distribution from a practical point of view is its symmetric behaviour. Azzalini (1985) provided a methodology to introduce skewness in a normal distribution and he defined the skew normal distribution as follows: A random variable X is said to have the skew normal distribution with skew- ness parameter λ ∈ R =(−∞, ∞), denoted by SND(λ), if its probability density function (p.d.f.) is of the following form, for x ∈ R. (1.1) g1(x; λ)=2f(x)F (λx) where f(·) and F (·) are respectively the p.d.f. and cumulative distribution func- tion (c.d.f.) of a standard normal variate. The SND(λ) has been studied by Azzalini (1985, 1986), Henze (1986) and several others. Further, various general- izations were studied by Azzalini and Dalla Valle (1996), Mudholkar and Hutson (2000), Branco and Dey (2001), Arnold and Beaver (2002), Arellano-Valle et al. (2004), Sharafi and Behboodian (2008), Jamalizadeh et al. (2008, 2009), and Kumar and Anusree (2011, 2013a, b, 2014a, b). Kim (2005) introduced a two-piece version of the SND(λ) through the fol- lowing p.d.f. 2πf(x)F (λ|x|) (1.2) g (x; λ)= . K (π + 2 tan−1(λ)) The distribution of a random variable with p.d.f. (1.2) hereafter we denote as TSNDK (λ). It is seen that there exists a symmetric behaviour in TSNDK (λ) Received August 12, 2014. Revised December 5, 2014. Accepted December 31, 2014. *Department of Statistics, University of Kerala, Trivandrum 695 581, India. Email: [email protected], [email protected] 180 C.SATHEESH KUMAR AND M.R.ANUSREE as determined by the sign of λ on either side of the origin. In order to mitigate this limitation, Kumar and Anusree (2013a) introduced a wide class of the two- piece skew normal distribution denoted by TSNDKA(λ, ρ), through the following p.d.f., in which ρ ∈ [−1, 1] and C =2π[π − tan−1(λ) + tan−1(ρλ)]−1. Cf(x)F (λx),x<0 (1.3) gKA(x; λ, ρ)= Cf(x)F (ρλx),x≥ 0. Note that TSNDKA(−λ, −1)isTSNDK (λ). In order to accomodate plurimodal- ity, Kumar and Anusree (2011) developed a generalized version of the SND(λ)by considering a mixture of the standard normal and SND(λ) through the following p.d.f., in which x ∈ R, λ ∈ R and α ≥−1. 2 (1.4) g (x; λ, α)= f(x)[1+ αF (λx)]. 2 (α +2) A distribution with p.d.f. (1.4) they notated as the GMNSND(λ, α). In order to develop a more flexible class of asymmetric, plurimodal normal type distribu- tions, through this paper we propose a two-piece version of the GMNSND(λ, α) and named it “the extended two-piece skew normal distribution” or in short “the ETPSND”. The ETPSND reduces to the standard normal distribution, the SND(λ) of Azzalini (1985), the GMNSND(λ, α), the two-piece skew normal dis- tribution of Kim (2005) and the generalized two-piece skew normal distribution of Kumar and Anusree (2013a) for particular choices of its shape parameters as its special cases. The rest of the paper is organized as follows: The definition of ETPSND and some of its important properties are given in Section 2. In Section 3, we obtain expressions for certain reliability measures and discuss also some concepts regarding the mode of the distribution. In Section 4, a location- scale extension of the ETPSND is suggested and some of its important structural properties are studied. Finally, the maximum likelihood estimation of the pa- rameters is discussed in Section 5 and its practical usefulness is illustrated with the help of certain real life data sets. We need the following notation in the sequel. For any a ∈ R and b>0, define ∞ bx (1.5) ξ(a, b)= f(x) f(y)dydx, a 0 so that tan−1(λ) (1.6) ξ(0,λ)= 2π and for any reals a, b and s such that bx + s>0, ∞ bx+s (1.7) ξs(a, b)= f(x) f(y)dydx. a 0 ON A GENERAL CLASS OF TWO-PIECE SKEW NORMAL DISTRIBUTION 181 2. Definition and properties First we present the definition of the extended two-piece skew normal distri- bution and discuss some of its properties. Definition 2.1. A random variable Z is said to follow the extended two- piece skew normal distribution with parameters λ1, λ2 ∈ R and α ≥−1 if its p.d.f. h(z; λ1,λ2,α) is of the followingform. For z ∈ R, 2 f(z)[1+ D(λ ,λ ,α)F (λ z)],z<0 (α +2) 1 2 1 (2.1) h(z; λ ,λ ,α)= 1 2 2 f(z)[1+ D(λ ,λ ,α)F (λ z)],z≥ 0 (α +2) 1 2 2 −1 −1 −1 where D(λ1,λ2,α)=απ[π + tan (λ2) − tan (λ1)] . Here D(λ1,λ2,α) is obtained by using Lemmas 2.1and 2.2 as given in Kumar and Anusree (2013a). The distribution of a random variable Z with p.d.f. (2.1) we denoted as ETPSND(λ1,λ2,α). For some particular choices of λ1, λ2 and α the p.d.f. given in (2.1) of ETPSND(λ1,λ2,α) is plotted and is shown in Fig. 1. Some important special cases of the ETPSND(λ1,λ2,α) are (i) ETPSND(λ1,λ2, 0), ETPSND(0, 0,α) or the limiting case of the ETPSND(λ1,λ2,α) when λ1 →−∞, λ2 →∞, is the standard normal distri- bution, (ii) the limiting case of the ETPSND(λ1,λ2,α) when α →∞is the TSNDKA(λ, ρ), (iii) the limiting case of the ETPSND(λ, λ, α) when α →∞is the TSNDK (λ), (iv) the ETPSND(λ, λ, α)istheGMNSND(λ, α), which reduces to SND(λ) when α = −1and Figure 1. Probability plots of ETPSND(λ1,λ2,α) for different choices of α. 182 C.SATHEESH KUMAR AND M.R.ANUSREE (v) the limiting case of the ETPSND(λ1,λ2,α)asλ1 →∞, λ2 →∞and α →∞or λ1 →−∞, λ2 →−∞and α →∞is the standard half normal distribution. We obtain certain structural properties of the ETPSND(λ1,λ2,α) through the following results. Result 2.1. If Z follows ETPSND(λ1,λ2,α) with p.d.f. h(z; λ1,λ2,α), then Y1 = −Z follows ETPSND(−λ2, −λ1,α). Proof. For any y1 ∈ R, the p.d.f. h1(y1; λ1,λ2,α)ofY1 is given by dz h1(y1; λ1,λ2,α)=h(−y1; λ1,λ2,α) dy1 2 f(y )[1+ D(λ ,λ ,α)F (−λ y )],y< 0 (α +2) 1 1 2 2 1 1 = 2 f(y )[1+ D(λ ,λ ,α)F (−λ y )],y≥ 0, (α +2) 1 1 2 1 1 1 which shows that Y1 = −Z follows ETPSND(−λ2, −λ1,α). Result 2.2. If Z follows ETPSND(λ1,λ2,α) with p.d.f. h(z; λ1,λ2,α), then 2 Y2 = Z has p.d.f. (2.2). Proof. The p.d.f. h2(y2; λ1,λ2,α)ofY2 is given by dz (2.2) h2(y2; λ1,λ2,α)=h(z; λ1,λ2,α) dy2 √ dz √ dz = h(− y2; λ1,λ2,α) + h( y2; λ1,λ2,α) dy dy √ 2 2 1 f( y ) = √ 2 (α +2) 2 y 2 √ √ × [2 + D(λ1,λ2,α)(F (−λ1 y2)+F (λ2 y2))]. Remark 2.1. When λ1 = λ2 and α = −1, the p.d.f. given in (2.2) reduces to the p.d.f. of a Chi-square variate with one degree of freedom. In order to find the distribution function of ETPSND(λ1,λ2,α), we need the following result. Result 2.3. If Z is a ETPSND(λ1,λ2,α), then for any real d1, d2 such that d1 ≤ d2, where ξ(a, b) is as given in (1.5). (2.3) P (d1 ≤ Z ≤ d2) ON A GENERAL CLASS OF TWO-PIECE SKEW NORMAL DISTRIBUTION 183 2 D(λ1,λ2,α) [F (d2) − F (d1)] + [F (d2) − F (d1)] (α +2) (α +2) D(λ ,λ ,α) + 1 2 [2ξ(d ,λ ) − 2ξ(d ,λ )],d≤ d < 0 (α +2) 1 1 2 1 1 2 = 2 D(λ1,λ2,α) [F (d ) − F (d )] + [F (d ) − F (d )] (α +2) 2 1 (α +2) 2 1 D(λ ,λ ,α) + 1 2 [2ξ(d ,λ ) − 2ξ(d ,λ )], 0 ≤ d ≤ d . (α +2) 1 2 2 2 1 2 Proof. For any d1 ≤ d2 < 0, by definition, d2 (2.4) P (d1 ≤ Z ≤ d2)= h(z; λ1,λ2,α)dz d1 d2 2 D(λ1,λ2,α) = f(z)+ 2f(z)F (λ1z) dz d1 (α +2) (α +2) 2 = [F (d ) − F (d )] (α +2) 2 1 D(λ ,λ ,α) + 1 2 [G(d ,λ ) − G(d ,λ )] (α +2) 2 1 1 1 where G(·,λ) is the distribution function of the SND(λ).
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