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 skewness 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 function (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 following 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 plurimodality, 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 distributions, 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 distribution 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 parameters 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 distribution 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 distribution, (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). Now, for the case 0 ≤ d1 ≤ d2, d2 (2.5) P (d1 ≤ Z ≤ d2)= h(z; λ1,λ2,α)dz d1 d2 2 D(λ1,λ2,α) = f(z)+ 2f(z)F (λ2z) 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 2 1 2 Thus (2.4) and (2.5) implies (2.3).

Result 2.4. The c.d.f. H(z) of a random variable Z following ETPSND(λ1,λ2,α) with p.d.f. (2.1) is the following, in which ξ(a, b) is as given in (1.5).   D(λ1,λ2,α) F (z)+ [F (z) − 2ξ(z,λ1)],z<0  2  2 D(λ1,λ2,α) (2.6) H(z)= F (z)+ F (z) − 2ξ(z,λ2) (α +2) 2   tan−1(λ ) tan−1(λ )  − 1 + 2 ,z≥ 0. π π 184 C.SATHEESH KUMAR AND M.R.ANUSREE

Proof. Let Z be a random variable with p.d.f. (2.1) and c.d.f. H(z). Then by deﬁnition, L ,z<0 (2.7) H(z)= 1 L2,z≥ 0, where z z 2 D(λ1,λ2,α) (2.8) L1 = f(t)dt+ 2f(t)F (λ1t)dt (α +2) −∞ (α +2) −∞ 2 D(λ ,λ ,α) = F (z)+ 1 2 G(z,λ ) (α +2) (α +2) 1 2 D(λ ,λ ,α) = F (z)+ 1 2 [F (z) − 2ξ(z,λ )] (α +2) (α +2) 1 and 0 2 (2.9) L = f(t)[1+ D(λ ,λ ,α)F (λ t)]dt 2 (α +2) 1 2 1 −∞ z 2 + f(t)[1+ D(λ1,λ2,α)F (λ2t)]dt 0 (α +2) 2 D(λ1,λ2,α) = F (z)+ G(0,λ1) (α +2) 2 D(λ, α, ρ) + (G(z,λ2) − G(0,λ2)) 2 2 D(λ1,λ2,α) 1 = F (z)+ − 2ξ(0,λ1) (α +2) 2 2 D(λ ,λ ,α) 1 + 1 2 F (z) − 2ξ(z,λ ) − − 2ξ(0,λ ) . 2 2 2 2

Since ξ(0, ·) is as deﬁned in (1.6), on substituting (2.8) and (2.9) in (2.7) we get (2.6).

Result 2.5. The characteristic function, φZ (t) of a random variable Z following ETPSND(λ1,λ2,α) with p.d.f. (2.1) is the following, for any t ∈ R and i2 = −1.

2 2 (2.10) φ (t)= e−t /2[1+ D(λ ,λ ,α)F (iδ t)] Z (α +2) 1 2 1

2 2 − D(λ ,λ ,α)e−t /2[ξ (−it, λ ) − ξ (−it, λ )] (α +2) 1 2 k1 1 k2 2

√λ1 where, δ1 = 2 for j =1, 2 and ξs(a, b) is deﬁned in (1.7). 1+λ1 ON A GENERAL CLASS OF TWO-PIECE SKEW NORMAL DISTRIBUTION 185

Proof. Let Z follow ETPSND(λ1,λ2,α) with p.d.f. (2.1). By the deﬁni- tion of the characteristic function, for any t ∈ R we have

itZ (2.11) φZ (t)=E(e ) ∞ itz = e h(z; λ1,λ2,α)dz. −∞

On substituting (2.1) in (2.11) and rearranging the terms in light of the expres- sion for the characteristic function of a standard normal variate, we obtain the following,

∞ 2 −t2/2 1 −(z−it)2/2 (2.12) φZ (t)= e 1+D(λ1,λ2,α) √ e F (λ1z)dz (α +2) −∞ 2π ∞ 1 −(z−it)2/2 − D(λ1,λ2,α) √ e F (λ1z)dz 0 2π ∞ 1 −(z−it)2/2 + D(λ1,λ2,α) √ e F (λ2z)dz . 0 2π If we put z − it = x in (2.12), we get ∞ −x2/2 2 −t2/2 e F (λ1(x + it))dx φZ (t)= e 1+D(λ1,λ2,α) √ (α +2) −∞ 2π ∞ −x2/2 e F (λ1(x + it))dx − D(λ1,λ2,α) √ − 2π it ∞ −x2/2 e F (λ2(x + it))dx + D(λ1,λ2,α) √ −it 2π 2 2 = e−t /2 1+D(λ ,λ ,α)F (iδ t) (α +2) 1 2 1 ∞ − D(λ1,λ2,α) f(x)F (λ1(x + it))dx −it ∞ + D(λ1,λ2,α) f(x)F (λ2(x + it))dx , −it in light of lemma 2.1of Kumar and Anusree (2013a). Now, on rearranging the terms and putting k = λit to arrive at the following.

2 2 φ (t)= e−t /2[1+ D(λ ,λ ,α)F (iδ t)] Z (α +2) 1 2 1 ∞ 0 λ1x+k1 D(2λ ,λ ,α) 2 − 1 2 e−t /2 f(x) f(u)du + f(u)du dx (α +2) − −∞ it 0 ∞ 0 λ2x+ρk2 2D(λ ,λ ,α) 2 + 1 2 e−t /2 f(x) f(u)du + f(u)du dx, (α +2) −it −∞ 0 which implies (2.10), in light of (1.7). 186 C.SATHEESH KUMAR AND M.R.ANUSREE

3. Reliability measures and mode

Here we present some reliability aspects of the ETPSND(λ1,λ2,α) with p.d.f. (2.1) and discuss some concepts regarding its mode.

Result 3.1. The reliability function R(t) of a random variable Z having the ETPSND(λ1,λ2,α) is the following, in which ξ(t, ·) is as deﬁned in (1.5).   D(λ ,λ ,α) F (t)+ 1 2 [F (t) − 2ξ(t, λ )],t<0  1  2  D(λ ,λ ,α) F (t)+ 1 2 [F (t) − 2ξ(t, λ )] 2 2 2 (3.1) R(t)=1−  D(λ ,λ ,α) (α +2) 1 2  +  2  −1 1  × tan−1(λ )+ tan−1(λ ) ,t≥ 0. π 1 π 2

Proof follows from the deﬁnition of the reliability function R(t)=1− H(t), where H(·) is the c.d.f. of Z as obtained in Result 2.4.

Result 3.2. The failure rate r(t) of a random variable Z following the ETPSND(λ1,λ2,α) with p.d.f. h(z; λ1,λ2,α)is   2f(t)[1 + D(λ1,λ2,α)F (λ1t)]  ,t<0  (α +2)− 2(F (t)+D(λ1,λ2,α)[F (t) − 2ξ(t, λ1)]) 

2f(t)[1 + D(λ1,λ2,α)F (λ2t)] (3.2) r(t)= ,  1 −1 1 −1  (α +2)− 2 F (t)+D(λ1,λ2,α) F (t) − 2ξ(t, λ2) − tan (λ1)+ tan (λ2)  π π  t ≥ 0.

h(t;λ1,λ2,α) Proof follows from the deﬁnition of failure rate, r(t)= R(t) , where R(t) is the reliability function as given in (3.1).

Result 3.3. The mean residual life function(MRLF) µ(t)of λ ETPSND(λ ,λ ,α) is the following, in which ∆ = 2 , δ = j for 1 2 (α+2)R(t) j 2 1+λj j =1.2.   δ1 f(t)+D(λ1,λ2,α)f(t)F (λ1t)+ √ D(λ1,λ2,α)  2 2π   δ1  − √ D(λ ,λ ,α)F (t 1+λ2)  1 2 1  2π δ + √2 D(λ ,λ ,α) − t, t ≤ 0 (3.3) µ(t)=∆ 1 2  2 2π   √δ2 f(t)+D(λ1,λ2,α)f(t)F (λ2t)+ D(λ1,λ2,α)  2 2π   δ2 2  − √ D(λ1,λ2,α)F (t 1+λ ) − t, t > 0 2π 2 ON A GENERAL CLASS OF TWO-PIECE SKEW NORMAL DISTRIBUTION 187

Proof. By deﬁnition, the mean residual life function of Z following the ETPSND(λ1,λ2,α) is given by (3.4) µ(t)=E(Z | Z>t) − t 2 = R(t)(α +2)   0  t zf(z)[1+ D(λ1,λ2,α)F (λ1z)]dz  ∞ + 0 zf(z)dz × ∞  + D(λ ,λ ,α) zf(z)F (λ z)dz − t, t ≤ 0  1 2 0 2  ∞ − t zf(z)[1+ D(λ1,λ2,α)F (λ2z)]dz t, t > 0. Now for any t<0, 0 (3.5) zf(z)[1+ D(λ1,λ2,α)F (λ1z)]dz t 0 = − f (z)[1+ D(λ1,λ2,α)F (λ1z)]dz t 1 1 = f(t) − √ − D(λ1,λ2,α) − F (λ1t)f(t) 2π 2 (2π) 0 − D(λ1,λ2,α)λ1 f(λ1z)f(z)dz t 1 D(λ1,λ2,α) = f(t) − √ − √ + D(λ1,λ2,α)F (λ1t)f(t) 2π 2 2π 1 λ D(λ ,λ ,α) − F (t 1+λ2) 1 1 2 2 1 + √ 2 2π 1+λ1 and for any t>0, ∞ (3.6) zf(z)[1+ D(λ1,λ2,α)F (λ2z)]dz t ∞ = − f (z)[1+ D(λ1,λ2,α)F (λ2z)]dz t λ2D(λ1,λ2,α) = f(t)+D(λ1,λ2,α)F (λ2t)f(t)+ √ 2π 1+λ2 2 λ − D(λ ,λ ,α)√ 2 F (t 1+λ2). 1 2 2 2 2π 1+λ2 In particular, when t = 0 in (3.6) we have ∞ 1 D(λ1,λ2,α) (3.7) zf(z)[1+ D(λ1,λ2,α)F (λ2z)]dz = + √ 0 (2π) 2 2π λ D(λ ,λ ,α) + √2 1 2 . 2 2 2π 1+λ2 188 C.SATHEESH KUMAR AND M.R.ANUSREE

Now on substituting (3.5), (3.6) and (3.7) in (3.4), we get (3.3).

Result 3.4. The p.d.f. of ETPSND(λ1,λ2,α) is bimodal with unimodes in the regions: z ∈ (−∞, 0) and z ∈ [0, ∞) subject to the conditions given below, in which for i =0, 1and j =1, 2,

i 1−i i D (λ1,λ2,α)(λjz) f(λjz) aij = i+1 . [1+ D(λ1,λ2,α)F (λjz)]

Region z ∈ (−∞, 0): (i) For all α ≥ 0 for which either λ1 ≤ 0orλ1 > 0 with |a01| 0 with 2 λ f(λ1z)(a01 + a11) > −1. Region z ∈ [0, ∞): (i) For all α ≥ 0 for which either λ2 > 0orλ2 ≤ 0 with |a02| 0 with |a12| 1. Proof. In order to show that there exists unimodes in regions of z ∈ (−∞, 0) and z ∈ [0, ∞), it is enough to show that the second derivative of h(z; λ1,λ2,α) is negative for all α, λ1 and λ2 in the respective regions. For z ∈ (−∞, 0), we have

d2 (3.8) {log[h(z; λ ,λ ,α)]} = −1 − λ2f(λ z)[a + a ] dz2 1 2 1 1 00 10 and for z ∈ [0, ∞), we have

d2 (3.9) {log[h(z; λ ,λ ,α)]} = −1 − λ2f(λ z)[a + a ]. dz2 1 2 2 2 01 11

Note that for all α,1+D(λ1,λ2,α)F (λjz) is positive for each j =1, 2. Now for the region z ∈ (−∞, 0), a01 is positive or negative according to whether the value of λ1 is negative or positive and a11 is positive or negative according to whether the value of α is positive or negative. If α ≥ 0, then (3.8) is negative either for λ1 ≤ 0 or for λ1 > 0ifa01 + a11 > 0 and for α<0, (3.8) is negative either 2 − for λ1 < 0 provided a01 + a11 > 0orforλ1 > 0 with λ1f(λ1z)(a01 + a11) > 1. Hence the p.d.f. given in (2.1) is log-concave and thus unimodal under these cases. Now for the region z ∈ [0, ∞), a02 is positive or negative according to whether the value of λ2 is positive or negative and a12 is positive or negative according to whether the value of α is positive or negative. If α ≥ 0, then (3.9) is negative either for λ2 > 0orforλ2 < 0ifa02 + a12 > 0. Now for α<0, (3.9) is negative 2 − either for λ2 > 0ifa02 + a12 > 0orλ2 < 0ifλ2f(λ2z)(a02 + a12) > 1. Thus the p.d.f. given in (2.1) is log-concave and hence unimodal under these conditions.

As a consequence of Result 3.4 we obtain the following result. ON A GENERAL CLASS OF TWO-PIECE SKEW NORMAL DISTRIBUTION 189

Result 3.5. The p.d.f. of ETPSND(λ1,λ2,α) is plurimodal in the regions: z ∈ (−∞, 0) and z ∈ [0, ∞) is as given below: Region z ∈ (−∞, 0): (i) For all α ≥ 0 and λ1 > 0 such that |a01| >a11. 2 (ii) For all α<0, λ1 ≤ 0 such that |a11| >a01 or λ1 > 0 and λ f(λ1z)(a01 + a11) < −1. Region z ∈ [0, ∞): (i) For all α ≥ 0, λ2 ≤ 0 such that |a02| >a12, (ii) For all α<0 for which either λ2 > 0 with |a12| >a02 or λ2 ≤ 0 with 2 − λ2f(λ2z)(a02 + a12) < 1.

4. Location-scale extension In this section we consider the location-scale extension of the ETPSND(λ1,λ2,α) and discuss some of its important properties. Let Z follow ETPSND(λ1,λ2,α), then X = µ + σZ is said to have a location scale extension of the two-piece skew normal distribution with location parameter µ, scale parameter σ and shape parameters λ1, λ2 and α, which is denoted as LS ETPSND (µ, σ; λ1,λ2,α). The p.d.f. of X is given by

(4.1) h(x; µ, σ, λ1,λ2,α)  2 x − µ   f (α +2)σ σ   x − µ  × 1+D(λ ,λ ,α)F λ ,x<µ 1 2 1 σ =  −  2 x µ  f (α +2)σ σ   x − µ  × 1+D(λ ,λ ,α)F λ ,x≥ µ, 1 2 2 σ in which µ, λ1, λ2 ∈ R, σ>0 and α ≥−1. Clearly we have the following special cases. LS LS (i) ETPSND (µ, σ; λ1,λ2, 0), ETPSND (µ, σ;0, 0,α) or the limiting case LS of the ETPSND (µ, σ; λ1,λ2,α) when λ1 →−∞, λ2 →∞, is the normal distribution with parameters µ and σ2, LS (ii) the limiting case of the ETPSND (µ, σ; λ1,λ2,α) when α →∞is the location scale extension of TSNDKA(λ, ρ), denoted by ETSNDKA(µ, σ; λ, ρ) (iii) the limiting case of the ETPSNDLS(µ, σ; λ, λ, α) when α →∞is the location scale extension of TSNDK (λ), denoted by ETSNDK (µ, σ; λ) (iv) the ETPSNDLS(µ, σ; λ, λ, α) is the location scale extension of GMNSND(λ, α), denoted by EGMNSND(µ, σ; λ, α) which reduces to the location scale extension of SND(λ), denoted by ESND(µ, σ; λ) when α = −1and LS (v) the limiting case of the ETPSND (µ, σ; λ1,λ2,α)asλ1 →∞, λ2 →∞ and α →∞or λ1 →−∞, λ2 →−∞and α →∞is the half normal distribution. LS We obtain the following properties of ETPSND (µ, σ; λ1,λ2,α) with p.d.f. (4.1) in a similar approach as we did in earlier sections. 190 C.SATHEESH KUMAR AND M.R.ANUSREE

Result 4.1. The characteristic function φX (t) of a random variable X fol- LS lowing ETPSND (µ, σ; λ1,λ2,α) is the following, in which kj = λjit, δj = λ j for j =1, 2. For i2 = −1and t ∈ R, 2 1+λj

2 2 2 φ (t)= eiµt−t σ /2[1+ D(λ ,λ ,α)F (iδ σt)] X (α +2) 1 2 1

2 2 2 − D(λ ,λ ,α)eiµt−t σ /2[ξ (−itσ, λ ) − ξ (−itσ, λ )]. (α +2) 1 2 k1σ 1 k2σ 2

Result ∗ 4.2. The distribution function H1 (t) of a random variable X fol- LS lowing ETPSND (µ, σ; λ1,λ2,α) is the following,  −  t µ D(λ1,λ2,α) F +  σ 2   t − µ t − µ  × F − 2ξ ,λ1 ,t<µ  σ σ  2 t − µ D(λ ,λ ,α) H∗(t)= F + 1 2 1 (α +2) σ 2   t − µ t − µ  × F − 2ξ ,λ  2  σ σ  − −  tan 1(λ ) tan 1(λ )  − 1 + 2 ,t≥ µ. π π

Result ∗ 4.3. The reliability function R1(t) of a random variable X following LS ETPSND (µ, σ; λ1,λ2,α) is the following,   t − µ D(λ ,λ ,α)  1 2 F +  σ 2   t − µ t − µ  × F − 2ξ ,λ1 ,t<µ  σ σ   t − µ D(λ ,λ ,α) F + 1 2 2 σ 2 R∗(t)=1− 1  t − µ t − µ (α +2) × −  F 2ξ ,λ2  σ σ   D(λ1,λ2,α)  −  2   −1 1  × tan−1(λ )+ tan−1(λ ) ,t≥ µ. π 1 π 2 ON A GENERAL CLASS OF TWO-PIECE SKEW NORMAL DISTRIBUTION 191

Result 4.4. The failure rate r1(t) of a random variable X following LS ETPSND (µ, σ; λ1,λ2,α) is the following, r1(t)  t−µ t−µ  2f σ 1+D(λ1,λ2,α)F λ1 σ  ,t<µ  σ(α +2)− 2σ F t−µ + D(λ ,λ ,α) F t−µ − 2ξ t−µ ,λ  σ 1 2 σ σ 1 t−µ t−µ = 2f σ 1+D(λ1,λ2,α)F λ2 σ  ,  t−µ t−µ t−µ 1 −1 1 −1  σ(α +2)− 2σ F + D(λ1,λ2,α) F − 2ξ ,λ2 − tan (λ1)+ tan (λ2)  σ σ σ π π t ≥ µ.

5. Estimation

Let X1,X2,...,Xn be a random sample of size n from LS ETPSND (µ, σ; λ1,λ2,α) with p.d.f. (4.1). Let X(1),X(2),...,X(n) be the or- dered sample. Assume X(r) <µlikelihood function l(θ)=l(µ, σ; λ1,λ2,α) of the sample is the following, { in which ΣIj , denotes the summation over the set Ij such that I1 = i : X(i) <µ, for i =1, 2,...,r} and I2 = {i : X(i) ≥ µ, for i = r +1,...,n}. 2 (5.1) l(θ)=n ln σ(α +2) x − µ λ (x − µ) + ln f i 1+D(λ ,λ ,α)F 1 i σ 1 2 σ I1 x − µ λ (x − µ) + ln f i 1+D(λ ,λ ,α)F 2 i . σ 1 2 σ I2

On diﬀerentiating (5.1) with respect to the parameters µ, σ, λ1, λ2 and α and equating to zero, we obtain the following likelihood equations, in which xi−µ d ui = for i =1, 2,...,n, D (λ1,λ2,α)= D(λ1,λ2,α) and for j =1, 2 and σ j dλk k 1−k xi−µ f (λj )F (λj ui) k =0, 1 βj(xi; λ1,λ2,α)=1+D(λ1,λ2,α)F (λi ), γjk(xi)= . σ βj (xi;λ1,λ2,α)

(x − µ) (x − µ) (5.2) i + i σ σ I1 I2     = D(λ1,λ2,α) λ1 γ11(xi)+λ2 γ21(xi) ,

I1 I2 n 1 (x − µ)2 1 (x − µ)2 (5.3) = i + i 2σ2 2 σ4 2 σ4 I1  I2  D(λ ,λ ,α) − 1 2 λ γ (x )(x − µ)+λ γ (x )(x − µ) , 2σ3 1 11 i i 2 21 i i I1 I2 192 C.SATHEESH KUMAR AND M.R.ANUSREE D(λ ,λ ,α) (5.4) 1 2 γ (x )(x − µ)+ γ (x )D (λ ,λ ,α) σ 11 i i 10 i 1 1 2 I1 I1 + D1(λ1,λ2,α)β1(xi; λ1,λ2,α)=0, I1 D(λ ,λ ,α) (5.5) 1 2 γ (x )(x − µ)+ γ (x )D (λ ,λ ,α) σ 21 i i 20 i 2 1 2 I1 I1 + D2(λ1,λ2,α)β2(xi; λ1,λ2,α)=0 I1 and nα (5.6) D(λ ,λ ,α) γ + D(λ ,λ ,α) γ = . 1 2 10 1 2 20 (α +2) I1 I2

Solving the non-linear system of equations (5.2) to (5.6) by simultaneous so- lution method using mathematical software such as MATHCAD, MATLAB, MATHEMATICA etc., the maximum likelihood estimates (MLE) of the param- LS eters of ETPSND (µ, σ; λ1,λ2,α) can be obtained.

6. Applications LS For illustrating the usefulness of the ETPSND (µ, σ; λ1,λ2,α), in this section we have considered the IQ data set for 52 non-white males and 87 white males hired by a large insurance company in 1971 given in Roberts (1988). Data set 1: 91 102 100 117 122 115 97 109 108 104 108 118 103 123 123 103 106 102 118 100 103 107 108 107 97 95 119 102 108 103 102 112 99 116 114 102 111 104 122 103 111 101 91 99 121 97 109 106 102 104 107 95. Data set 2: 85 94 94 97 98 100 100 101 102 102 103 103 103 103 104 104 106 106 106 106 106 107 107 108 108 108 108 108 108 108 109 109 111 111 112 112112112112112112112112112113113113113113113113113114114 115 116 116 116 116 117 117 117 118 118 118 119 120 120 120 121 121 121 122 122 122 122 122 122 124 124 125 129 131 132 135 136 140. Here we have ﬁtted the proposed model to these two data sets, non-white males and white males, and compared results with the existing models such as the normal distribution N(µ, σ), ESND(µ, σ, λ), ETSNDK (µ, σ; λ, ρ) and LS ETPSND (µ, σ; λ1,λ2,α). For model comparison, we have computed some well- known information criterion-such as Akaike’s Information Criterion (AIC), the Bayesian Information Criterion (BIC) and the corrected Akaike’s Information Criterion (AICc). For numerical evaluation, we have used MATHCAD software and the results obtained are included in Table 1for the case of Data set 1and in Table 2 for the case of Data set 2 along with the log-likelihood(l). In light of the computed values listed in Tables 1and 2, it can be seen that LS the ETPSND (µ, σ; λ1,λ2,α) gives the best results compared to other existing models. ON A GENERAL CLASS OF TWO-PIECE SKEW NORMAL DISTRIBUTION 193

LS Table 1. MLE of the parameters of the ETPSND (µ, σ; λ1,λ2,α) and computed values of the l, AIC, BIC and AICc values in the case of various ﬁtted models to data set 1.

LS N ESND ETSND K ETPSND

Distribution: (µ, σ) (µ, σ, λ) (µ, σ; λ, ρ) (µ, σ; λ1,λ2,α) µˆ 106.653 98.79 104.64 102.45 σˆ 8.229 11.38 8.9 9.8 λˆ — 1.71 0.8 2.54, −0.457 αˆ — — — 1.66 ρˆ — — −0.04 — l −183.374 −182.423 −182 −177 AIC 371 371 370 365 BIC 375 377 376 374 AICc 370.99 371.34 370.73 366

LS Table 2. MLE of the parameters of the ETPSND (µ, σ; λ1,λ2,α) and computed values of the l, AIC, BIC and AICc values in the case of various ﬁtted models to data set 2.

LS N ESND ETSND K ETPSND

Distribution: (µ, σ) (µ, σ, λ) (µ, σ; λ, ρ) (µ, σ; λ1,λ2,α) µˆ 112.86 105.78 107.1 108.3 σˆ 9.58 11.94 10.9 11.5 λˆ — 1.14 0.899 0.49, −0.0098 αˆ — — — −0.91 ρˆ — — 0.9 — l −319.6 −319.29 −319 −314 AIC 643.2 644.57 647 638 BIC 648.14 651.97 657 647 AICc 643.35 644.86 647.32 639.01

Acknowledgements The authors are grateful to the Editor, Associate Editor and the anony- mous referee for carefully reading the paper and for the valuable comments and suggestions. The second author is thankful to the Department of Science and Technology, Ministry of Science and Technology, New Delhi for ﬁnancial support (IF 110142).

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