A generalized skew two-piece skew-normal distribution - Springer Link

Stat Papers (2011) 52:431–446 DOI 10.1007/s00362-009-0240-x REGULAR ARTICLE

A generalized skew two-piece skew-normal distribution A. Jamalizadeh · A. R. Arabpour · N. Balakrishnan

Received: 16 March 2009 / Revised: 31 May 2009 / Published online: 19 June 2009 © Springer-Verlag 2009

Abstract In this paper, we discuss a general class of skew two-piece skew-normal distributions, denoted by GSTPSN (λ1 , λ2 , ρ). We derive its moment generating function and discuss some simple and interesting properties of this distribution. We then discuss the modes of these distributions and present a useful representation theorem as well. Next, we focus on a different generalization of the two-piece skew-normal distribution which is a symmetric family of distributions and discuss some of its properties. Finally, three well-known examples are used to illustrate the practical usefulness of this family of distributions. Keywords Skew-normal distribution · Generalized skew-normal distribution · Two-piece skew-normal distribution · Generalized two-piece skew-normal distribution · Generalized skew two-piece skew-normal distribution · Unimodality and bimodality · Orthant probability 1 Introduction A random variable Z λ is said to have a standard skew-normal distribution with parameter λ ∈ R, denoted by Z λ ∼ S N (λ), if its probability density function (pdf) is φ S N (z; λ) = 2φ(z)(λz), z ∈ R, λ ∈ R,

(1)

A. Jamalizadeh · A. R. Arabpour Shahid Bahonar University of Kerman, Kerman, Iran e-mail: [email protected] N. Balakrishnan (B) McMaster University, Hamilton, ON, Canada e-mail: [email protected]

123

432

A. Jamalizadeh et al.

where φ(z) and (z) denote the standard normal pdf and cumulative distribution function (cdf), respectively. This distribution and its variations have been discussed by several authors including Azzalini (1985, 1986), Henze (1986), Azzalini and Dalla Valle (1996), Branco and Dey (2001), Loperfido (2001), Arnold and Beaver (2002), Balakrishnan (2002), and Azzalini and Chiogna (2004). A comprehensive survey of developments on skew-normal distribution and its multivariate form is due to Azzalini (2005). Recently, Kim (2005) presented a symmetric two-piece skew-normal distribution. A random variable Z λ∗ is said to have a two-piece skew-normal distribution with parameter λ ∈ R, denoted by Z λ∗ ∼ TPSN (λ), if its pdf is ∗ φTPSN (z; λ) =

2π φ(z)(λ|z|), z ∈ R, λ ∈ R. π + 2 tan−1 (λ)

(2)

In the special case when λ = 0, then both pdf’s in (1) and (2) reduce to the standard normal pdf. Kim (2005) showed that the two-piece skew-normal density in (2) is uni/bimodal and is, in fact, a mixture of two truncated skew-normal distributions. Specifically, if we truncate the skew-normal random variable Z λ ∼ S N (λ) in (1) from above at zero and from below at zero and denote them by Z λ− ∼ S N − (λ) and Z λ+ ∼ S N + (λ), respectively, then the corresponding pdf’s of Z λ− and Z λ+ are given by 4π φ (z) (λz), z < 0, λ ∈ R, π − 2 tan−1 (λ) 4π φ (z) (λz), z > 0, λ ∈ R; (z; λ) = π + 2 tan−1 (λ)

φ S−N (z; λ) =

(3)

φ S+N

(4)

then, we have ∗TPSN (z; λ) =

1 − S N (z; −λ) + + S N (z; λ) , 2

+ where ∗TPSN (z; λ) denotes the cdf of Z λ∗ ∼ TPSN (λ), while − S N (z; λ) and S N (z; λ) denote the cdf’s of the truncated skew-normal distributions in (3) and (4), respectively. In this paper, we present a three-parameter generalized skew two-piece skew-normal distribution, denoted by GSTPSN (λ1 , λ2 , ρ), through a standard bivariate normal distribution with correlation ρ. This includes as special cases the Azzalini skew-normal distribution in (1) and the two-piece skew-normal distribution in (2). The rest of this paper is organized as follows. In Sect. 2, we describe this generalized skew two-piece skew-normal distribution and discuss some of its simple and interesting properties. In Sect. 3, we derive the moment generating function of GSTPSN (λ1 , λ2 , ρ) in an explicit form. In Sect. 4, we discuss the modes of this distribution and also present a useful representation theorem. Next, in Sect. 5, we focus on a different generalization of the two-piece skew-normal distribution, which is a symmetric family of distributions, and discuss some of its properties. Finally, in Sect. 6, we use three well-known examples to illustrate the practical usefulness of this family of distributions.

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A generalized skew two-piece skew-normal distribution

433

2 A generalized skew two-piece skew-normal distribution Definition 1 A random variable Z λ∗1 ,λ2 ,ρ is said to have a generalized skew two-piece skew-normal distribution, denoted by Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (λ1 , λ2 , ρ), with parameters λ1 , λ2 ∈ R and |ρ| < 1, if Z λ∗1 ,λ2 ,ρ = X | (Y1 < λ1 X, Y2 < λ2 |X |) d

(5)

where X ∼ N (0, 1) independently of (Y1 , Y2 )T ∼ N2 (0, 0, 1, 1, ρ). After some simple algebraic calculations, we can show that the pdf of Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (λ1 , λ2 , ρ) is ∗ φGSTPSN (z; λ1 , λ2 , ρ) = c∗ (λ1 , λ2 , ρ)φ(z)2 (λ1 z, λ2 |z|; ρ), z ∈ R,

(6)

where 2 (· , · ; ρ) denote the cdf of N2 (0, 0, 1, 1, ρ), and c∗ (λ1 , λ2 , ρ) =

1 a ∗ (λ1 , λ2 , ρ)

(7)

with a ∗ (λ1 , λ2 , ρ) = Pr (Y1 < λ1 X, Y2 < λ2 |X |) .

(8)

Lemma 1 We have ⎧ ⎛ ⎞ ⎛ ⎞ 1 ⎨ −1 ⎝ −(ρ + λ1 λ2 ) ⎠ ∗ −1 ⎝ −(ρ − λ1 λ2 ) ⎠ a (λ1 , λ2 , ρ) = + cos cos 4π ⎩ 1 + λ21 1 + λ22 1 + λ21 1 + λ22 ⎫ ⎬ + 2 tan−1 (λ2 ) . (9) ⎭ Proof Consider a ∗ (λ1 , λ2 , ρ) = Pr (Y1 < λ1 X, Y2 < λ2 |X |) = Pr (Y1 < λ1 X, Y2 < λ2 |X |, X > 0) + Pr (Y1 < λ1 X, Y2 < λ2 |X |, X ≤ 0) = Pr (Y1 − λ1 X < 0, Y2 − λ2 X < 0, −X < 0) + Pr (Y1 − λ1 X < 0, Y2 + λ2 X < 0, X ≤ 0) .

(10)

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434


By using the orthant probability expression for

Y 1 −λ1 X Y , 2 −λ2 X2 , −X 1+λ21 1+λ2

[see, for

example, Kotz et al. (2000)], we obtain Pr (Y1 − λ1 X < 0, Y2 − λ2 X < 0, −X < 0) ⎫ ⎧ ⎛ ⎞ ⎬ ⎨ −(ρ + λ1 λ2 ) ⎠ 1 = + tan−1 (λ1 ) + tan−1 (λ2 ) , cos−1 ⎝ ⎭ 4π ⎩ 1 + λ21 1 + λ22

(11)

and similarly Pr (Y1 − λ1 X < 0, Y2 + λ2 X < 0, X ≤ 0) ⎫ ⎧ ⎛ ⎞ ⎬ 1 ⎨ −1 ⎝ −(ρ − λ1 λ2 ) ⎠ = − tan−1 (λ1 ) + tan−1 (λ2 ) . cos ⎭ 4π ⎩ 1 + λ21 1 + λ22

(12)

Now, the Lemma follows readily from Eqs. (10) to (12). Thus, c∗ (λ1 , λ2 , ρ) =

cos−1

−(ρ+λ 1 λ2 ) 1+λ21 1+λ22

4π + cos−1

−(ρ−λ 1 λ2 ) 1+λ21 1+λ22

, (13) +2 tan−1 (λ

2)

and consequently the generalized skew two-piece skew normal pdf in (6) becomes, for z ∈ R, ∗ φGSTPSN (z; λ1 , λ2 , ρ)

= cos−1

4π

−(ρ+λ 1 λ2 ) 1+λ21 1+λ22

+ cos−1

×φ(z)2 (λ1 z, λ2 |z|; ρ).

−(ρ−λ 1 λ2 ) 1+λ21 1+λ22

+ 2 tan−1 (λ2 ) (14)

In Fig. 1, we have presented the plot of the generalized skew two-piece skew normal density in (14) for some choices of the parameters λ1 , λ2 and ρ in order to display the different shapes that this density could take on. In the special case when ρ = 0, we obtain a skew two-piece skew-normal distribution, denoted by Z λ∗1 ,λ2 ∼ STPSN (λ1 , λ2 ), with pdf ∗ φSTPSN (z; λ1 , λ2 ) =

123

4π φ(z)(λ1 z)(λ2 |z|), λ1 , λ2 , z ∈ R. (15) π + 2 tan−1 (λ2 )


435

φ*GSTPSN (z,0.1,3,0.5)

0.0

0.0

0.1

0.5

0.2

1.0

0.3

1.5

φ*GSTPSN (z,0.1,−3,0.5)

−3

−2

−1

0

1

2

3

−3

φ*GSTPSN (z,0.1,3,−0.5)

−2

−1

0

1

2

3

2

3

0.0

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

φ*GSTPSN (z,−0.1,3,0.5)

−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

Fig. 1 Plots of the generalized skew two-piece skew normal density for some choices of the parameters λ1 , λ2 and ρ ∗ It should be noted that φSTPSN (z; λ1 , λ2 ) in (15) is an example of the skew-symmetric distributions discussed by Wang et al. (2004) [see also Azzalini (2005)], since ∗ ∗ (z; λ1 , λ2 ) = 2φTPSN (z; λ2 )(λ1 z), φSTPSN

(16)

∗ (x; λ) is the two-piece skew normal density in (2). where φTPSN Recently, Jamalizadeh and Balakrishnan (2008, 2009) discussed a three-parameter generalized skew-normal distribution which includes the skew-normal distribution in (1) as a special case. A random variable Z λ1 ,λ2 ,ρ is said to have a generalized skew-normal distribution with parameters λ1 , λ2 ∈ R and |ρ| < 1, denoted by Z λ1 ,λ2 ,ρ ∼ G S N (λ1 , λ2 , ρ), if d

Z λ1 ,λ2 ,ρ = X | (Y1 < λ1 X, Y2 < λ2 X ), where X ∼ N (0, 1) independently of (Y1 , Y2 )T ∼ N2 (0, 0, 1, 1, ρ) as before. Then, the pdf of Z λ1 ,λ2 ,ρ is [see Jamalizadeh and Balakrishnan (2008, 2009)]

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436


φG S N (z; λ1 , λ2 , ρ) = cos−1

2π

φ (z) 2 (λ1 z, λ2 z; ρ) .

(17)

−(ρ+λ 1 λ2 ) 1+λ21 1+λ22

If we truncate the random variable Z λ1 ,λ2 ,ρ ∼ GSN (λ1 , λ2 , ρ) from above at zero and from below at zero and denote them by Z λ−1 ,λ2 ,ρ ∼ GSN − (λ1 , λ2 , ρ) and Z λ+1 ,λ2 ,ρ ∼ GSN + (λ1 , λ2 , ρ), respectively, then the corresponding pdf’s are − φGSN (z; λ1 , λ2 , ρ) = c− (λ1 , λ2 , ρ) φ (z) 2 (λ1 z, λ2 z; ρ) , + φGSN (z; λ1 , λ2 , ρ) = c+ (λ1 , λ2 , ρ) φ (z) 2 (λ1 z, λ2 z; ρ) ,

z < 0, z > 0,

(18) (19)

,

(20)

.

(21)

where c− (λ1 , λ2 , ρ) =

−(ρ+λ 1 λ2 ) 1+λ21 1+λ22

cos−1 c+ (λ1 , λ2 , ρ) =

cos−1

− tan−1 (λ

−(ρ+λ 1 λ2 ) 1+λ21 1+λ22

4π 1

) − tan−1 (λ

1

) + tan−1 (λ

2)

4π + tan−1 (λ

2)

Then, the following theorem shows that the generalized skew two-piece skew-normal distribution in (14) is, in fact, a mixture of the truncated distributions in (18) and (19), which is the reason that we call it a “generalized skew two-piece skew-normal distribution”. Theorem 1 If ∗ (z; λ1 , λ2 , ρ) denotes the cdf of Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (λ1 , λ2 , ρ), then + ∗ (z; λ1 , λ2 , ρ) = ω− GSN (z; λ1 , −λ2 , ρ) + (1 − ω) GSN (z; λ1 , λ2 , ρ) , z ∈ R, (22) c∗ (λ1 ,λ2 ,ρ) c− (λ1 ,−λ2 ,ρ)

with c∗ (λ1 , λ2 , ρ) and c− (λ1 , λ2 , ρ) being as in (13) and − (20), respectively, and GSN (z; λ1 , −λ2 , ρ) and + GSN (z; λ1 , λ2 , ρ) denote the cdf’s of Z λ−1 ,λ2 ,ρ ∼ GSN − (λ1 , −λ2 , ρ) and Z λ+1 ,λ2 ,ρ ∼ GSN + (λ1 , λ2 , ρ). where ω =

Some simple properties We now mention some simple properties of the generalized skew two-piece skew-normal density in (14): (1) GSTPSN (0, 0, ρ) = N (0, 1), |ρ| < 1 , GSTPSN (λ, 0, 0) = SN (λ) and GSTPSN (0, λ, 0) = TPSN (λ); (2) If Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (λ1 , λ2 , ρ) ⇐⇒ −Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (−λ1 , λ2 , ρ); d

(3) If Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (λ1 , λ2 , ρ), then Z λ∗1 ,λ2 ,ρ −→ Z λ1 when λ2 → ∞, where Z λ1 ∼ S N (λ1 );

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437

(4) Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (λ1 , λ2 , ρ), then Z λ∗1 ,λ2 ,ρ −→ Z λ+2 when λ1 → ∞, where Z λ+2 ∼ S N + (λ2 ); d (5) If Z λ∗1 ,λ2 ∼ STPSN (λ1 , λ2 ) and Z λ∗2 ∼ TPSN (λ2 ), then h Z λ∗1 ,λ2 = h Z λ∗2 for any even function h. d

3 Moment generating function of GSTPSN Theorem 2 The moment generating function of Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (λ1 , λ2 , ρ) is 1 2

MGSTPSN (s; λ1 , λ2 , ρ) = c∗ (λ1 , λ2 , ρ)e 2 s {3 (λ1 s, −λ2 s, −s; 1 ) + 3 (λ1 s, λ2 s, s; 2 )},

(23)

where c∗ (λ1 , λ2 , ρ) is as in (13), 3 (· , · , · ; ) denotes the cdf of N3 (0, ) and ⎛

⎛ ⎞ ⎞ 1 + λ21 ρ − λ1 λ2 −λ1 1 + λ21 ρ + λ1 λ2 λ1 1 = ⎝ ρ − λ1 λ2 1 + λ22 λ2 ⎠ and 2 = ⎝ ρ + λ1 λ2 1 + λ22 λ2 ⎠ . (24) −λ1 λ2 1 λ1 λ2 1 Proof From (14), we have the MGF of Z λ∗1 ,λ2 ,ρ as +∞

∗

MGSTPSN (s; λ1 , λ2 , ρ) = c (λ1 , λ2 , ρ)

esz φ (z) 2 (λ1 z, λ2 |z|; ρ) dz

−∞

= c∗ (λ1 , λ2 , ρ) e

1 2 2s

⎧ 0 ⎨ ⎩

φ (z − s) 2 (λ1 z, −λ2 z; ρ) dz

−∞

⎫ +∞ ⎬ + φ (z − s) 2 (λ1 z, λ2 z; ρ) dz ⎭ 0 ∗

1 2

= c (λ1 , λ2 , ρ) e 2 s {Pr (Y1 < λ1 U, Y2 < −λ2 U, U < 0) + Pr (Y1 < λ1 U, Y2 < λ2 U, U > 0)}, where U ∼ N (s, 1) and is independent of (Y1 , Y2 )T ∼ N2 (0, 0, 1, 1, ρ). Now, since d

U = X + s, where X ∼ N (0, 1), we immediately obtain MGSTPSN (s; λ1 , λ2 , ρ) 1 2

= c∗ (λ1 , λ2 , ρ)e 2 s {Pr (Y1 − λ1 X < λ1 s, Y2 + λ2 X < −λ2 s, X < −s)

123

438


+ Pr (Y1 − λ1 X < λ1 s, Y2 − λ2 X < λ2 s, −X < s)} 1 2

= c∗ (λ1 , λ2 , ρ)e 2 s {3 (λ1 s, −λ2 s, −s; 1 ) + 3 (λ1 s, λ2 s, s; 2 )} , which completes the proof of the theorem.

We can obtain the moments of Z λ∗1 ,λ2 ,ρ ∼ GSTPSN (λ1 , λ2 , ρ) readily from the derivatives of MGSTPSN (s; λ1 , λ2 , ρ) in (23). For example, we get the first two moments as ⎧ ⎡ ⎛ ⎞ ⎨ ∗ (λ , λ , ρ) λ λ c − λ ρ 1 2 2 1 ⎣ 1 ⎠ E Z λ∗1 ,λ2 ,ρ = π + tan−1 ⎝ 3 ⎩ 2 2 2 (2π ) 2 1 + λ1 1 − ρ 1 + λ1 ⎛ ⎞⎫ ⎬ λ2 + λ1 ρ ⎠ + tan−1 ⎝ ⎭ 1 + λ21 1 − ρ 2 ⎧ ⎛ ⎞ ⎨ λ λ2 − λ ρ 1 2 ⎠ + π + tan−1 ⎝ ⎩ 2 2 2 1 + λ2 1 − ρ 1 + λ2 ⎛ ⎞⎫⎤ ⎬ λ1 + λ2 ρ ⎠ ⎦, + tan−1 ⎝ (25) ⎭ 1 + λ22 1 − ρ 2 ⎧ 2 ∗ (λ , λ , ρ) ⎨ 2λ c 1 2 2 E Z λ∗1 ,λ2 ,ρ = 1 + ⎩ 1 + λ22 4π λ1 (λ2 −λ1 ρ) λ2 (λ1 −λ2 ρ) 1 + + 1 + λ21 1+λ22 1 − ρ 2 + λ21 + λ22 − 2ρλ1 λ2 ⎫ 1 λ1 (λ2 +λ1 ρ) λ2 (λ1 +λ2 ρ) ⎬ − + . ⎭ 1 + λ21 1+λ22 1−ρ 2 +λ21 +λ22 −2ρλ1 λ2 (26) 4 Modes of GSTPN and a representation theorem Kim (2005) showed that the two-piece skew-normal density in (2) is uni/bimodal. The following theorem shows that the generalized skew two-piece skew-normal density in (14) is also uni/bimodal, which is thus more flexible than the two-piece skewnormal distribution, as it is asymmetric and also includes the two-piece skew-normal distribution as a special case. ∗ (z; λ1 , λ2 , ρ) Theorem 3 The generalized skew two-piece skew-normal pdf φGSTPSN in (14) is uni/bimodal. Specifically, it is unimodal if λ2 ≤ 0 and it is bimodal if λ2 > 0.

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439

Kim (2005) showed that if (X 1 , X 2 )T ∼ N2 (0, 0, 1, 1, ρ12 ), then: 1. X 1 | (X 1 > 0, X 2 > 0) ∼ S N +

;

ρ12 2 1−ρ12

2. X 1 | (X 1 < 0, X 2 > 0) ∼ S N − 3. X 1 | (X 1 X 2 > 0) ∼ TPSN

ρ12 2 1−ρ12

;

ρ12 2 1−ρ12

.

In the following theorem, we present a generalization of these results. Theorem 4 (A representation theorem). Let (X 1 , X 2 , X 3 )T ∼ N3 (0, R), where R is a positive definite correlation matrix given by ⎛

⎞ 1 ρ12 ρ13 R = ⎝ ρ12 1 ρ23 ⎠ ρ13 ρ23 1 and ρ23.1 =

ρ23 − ρ12 ρ13 . 2 2 1 − ρ12 1 − ρ13

Then:

(a) X 1 | (X 1 > 0, X 2 > 0, X 3 > 0) ∼

ρ12

GSN +

2 1−ρ12

(b) X 1 | (X 1 < 0, X 2 > 0, X 3 > 0) ∼ GSN − (c) X 1 | (X 1 > 0, X 2 > 0, X 3 < 0) ∼

GSN +

(d) X 1 | (X 1 < 0, X 2 > 0, X 3 < 0) ∼

GSN −

,

ρ13

2 1−ρ13

, ρ23.1 ;

ρ12 , ρ13 , ρ23.1 2 2 1−ρ12 1−ρ13 ρ12

2 1−ρ12

,−

ρ13

2 1−ρ13

;

, −ρ23.1 ;

ρ12 , − ρ13 , −ρ23.1 2 2 1−ρ12 1−ρ13

;

d

(e) If U = X 1 | (X 1 X 3 > 0, X 2 > 0) and FU (· ; R) denotes the cdf of U , then ⎛ FU (u; R) = ω∗ − ⎝u;

⎞ ρ12 2 1 − ρ12

⎛

ρ13

,− , −ρ23.1 ⎠ 2 1 − ρ13

+(1 − ω∗ )+ ⎝u;

ρ12 2 1 − ρ12

,

ρ13 2 1 − ρ13

⎞

, ρ23.1 ⎠ ,

123

440


where − (· ; λ1 , λ2 , ρ) and + (· ; λ1 , λ2 , ρ) denote the cdf’s of Z λ−1 ,λ2 ,ρ ∼ GSN − (λ1 , λ2 , ρ) and Z λ+1 ,λ2 ,ρ ∼ GSN + (λ1 , λ2 , ρ), respectively, and ∗

2 1−ρ12

ω = c−

ρ12

2 1−ρ12

,−

ρ13

ρ12

c+

2 1−ρ13

,

ρ13

, ρ23.1

2 1−ρ13

; ρ12

, −ρ23.1 + c+

2 1−ρ12

,

ρ13

2 1−ρ13

, ρ23.1

(f) If ρ23 = ρ12 ρ13 we have ⎛

⎞

ρ12 ρ13 ⎠ d U = X 1 | (X 1 X 3 > 0, X 2 > 0) ∼ STPSN ⎝ , . 2 2 1 − ρ12 1 − ρ13

(27)

Proof Proofs of (a–d) are straightforward since ⎛

⎞ ρ12

ρ13

, , ρ23.1 ⎠ ; X 1 | (X 2 > 0, X 3 > 0) ∼ GSN ⎝ 2 2 1 − ρ12 1 − ρ13 see Jamalizadeh and Balakrishnan (2008, 2009). For proving (e), we note that FU (u; R) = Pr (X 1 ≤ u | X 1 X 3 > 0, X 2 > 0) = Pr (X 1 < 0 | X 1 X 3 > 0, X 2 > 0) Pr (X 1 ≤ u | X 1 < 0, X 3 < 0, X 2 > 0) + Pr (X 1 > 0 | X 1 X 3 > 0, X 2 > 0) Pr (X 1 ≤ u | X 1 > 0, X 3 > 0, X 2 > 0) . Now, upon using (d) and (a) and the fact that P (X 1 < 0 | X 1 X 3 > 0, X 2 > 0) c+

= c−

ρ12

2 1−ρ12

,−

ρ13

2 1−ρ13

ρ12

2 1−ρ12

,

, −ρ23.1

ρ13

, ρ23.1

2 1−ρ13

+ c+

, ρ12

2 1−ρ12

,

ρ13

2 1−ρ13

, ρ23.1

we obtain (e). Now, by using (e) with ρ23.1 = 0 and Theorem 1, we readily obtain (f), and this completes the proof of the theorem. 5 A generalized two-piece skew-normal distribution In this section, we focus on a different generalization of the two-piece skew-normal distribution which is a symmetric family of distributions and includes the two-piece skew-normal distribution in (2) as a special case. This family should be contrasted

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441

with the generalized skew two-piece skew-normal distribution discussed earlier in Sect. 2 which is in general a skewed family of distributions and includes the skewnormal distribution in (1) and (2) as special cases. However, it is important to mention here that both these families of distributions are indeed mixtures of the generalized skew-normal distributions presented earlier in (17). Specifically, a random variable Z λ∗∗1 ,λ2 ,ρ is said to have a generalized two-piece skew-normal distribution, denoted by Z λ∗∗1 ,λ2 ,ρ ∼ GTPSN (λ1 , λ2 , ρ) with parameters λ1 , λ2 ∈ R and |ρ| < 1, if Z λ∗∗1 ,λ2 ,ρ = X | (Y1 < λ1 |X | , Y2 < λ2 |X |), d

where X ∼ N (0, 1) independently of (Y1 , Y2 )T ∼ N2 (0, 0, 1, 1, ρ) as before. We can then easily show that the pdf of Z λ∗∗1 ,λ2 ,ρ ∼ GTPSN (λ1 , λ2 , ρ) is given by ∗∗ φGTPSN (z; λ1 , λ2 , ρ) = c∗∗ (λ1 , λ2 , ρ) φ (z) 2 (λ1 |z| , λ2 |z|; ρ), z ∈ R

with c∗∗ (λ1 , λ2 , ρ) =

cos−1

−(ρ+λ 1 λ2 ) 1+λ21 1+λ22

2π

.

+ tan−1 (λ

1

) + tan−1 (λ

2)

This distribution is symmetric and it includes the two-piece skew-normal distribution in (2) as a special case. In the following theorem, we list some properties of Z λ∗∗1 ,λ2 ,ρ ∼ GTPSN (λ1 , λ2 , ρ). ∗∗ Theorem 5 (a) If ∗∗ GTPSN (z; λ1 , λ2 , ρ) denotes the cdf of Z λ1 ,λ2 ,ρ ∼ GTPSN (λ1 , λ2 , ρ), then

∗∗ GTPSN (z; λ1 , λ2 , ρ)=

1 − (z; −λ1 , −λ2 , ρ) + + (z; λ1 , λ2 , ρ) , z ∈ R, 2

where − (· ; −λ1 , −λ2 , ρ) and + (· ; λ1 , λ2 , ρ) denote the cdf’s of GSN − (−λ1 , −λ2 , ρ) and GSN + (λ1 , λ2 , ρ), respectively, as defined before; (b) If (X 1 , X 2 , X 3 )T ∼ N3 (0, R) with R as given in Theorem 4, we have X 1 | (X 1 X 2 > 0, X 1 X 3 > 0)

⎛

=X 1 | (X i have the same sign) ∼ GTPSN ⎝ d

⎞ ρ12 2 1−ρ12

ρ13

, , ρ23.1 ⎠ ; 2 1−ρ13

(c) The moment generating function of Z λ∗∗1 ,λ2 ,ρ ∼ GTPSN (λ1 , λ2 , ρ) is 1 2

MGTPSN (s; λ1 , λ2 , ρ) = c∗∗ (λ1 , λ2 , ρ) e 2 s {3 (−λ1 s, −λ2 s, −s; 2 ) + 3 (λ1 s, λ2 s, s; 2 )} ,

123

442


where 3 (· , · , · ; ) is the cdf of N3 (0, ) and 2 is as given in Theorem 2; (d) Since Z λ∗∗1 ,λ2 ,ρ ∼ GTPSN (λ1 , λ2 , ρ) is a symmetric distribution, all the odd moments are zero. In addition, we have E

2 Z λ∗∗1 ,λ2 ,ρ

c∗∗ (λ1 , λ2 , ρ) = 1+ 2π +

λ1 λ2 + 2 1 + λ1 1 + λ22

1 1 − ρ 2 + λ21 + λ22 − 2ρλ1 λ2

λ1 (λ2 − λ1 ρ) λ2 (λ1 − λ2 ρ) × + 1 + λ21 1 + λ22

.

6 Illustrative examples To illustrate the applicability of the GSTPSN model introduced in the preceding sections, we consider three real data sets. The first data set give the heights (in centimeters) of 100 Australian female athletes, the second data set consist of the hemoglobin levels of 202 Australian athletes, and the third data set gives the Otis IQ Scores of 52 nonwhite males. It needs to be mentioned that the first two data sets are related to the AIS (Australian Institute of Sport) data, which have been used extensively in the literature and are also available in the sn package of the R-software, have been analyzed earlier by Cook and Weisberg (1994). By performing a location-scale transformation, we first of all introduce the random ∗ ∗ 2 variable X ξ,ω 2 ,λ ,λ ,ρ = ξ + ω Z λ1 ,λ2 ,ρ ∼ GSTPSN (ξ, ω , λ1 , λ2 , ρ) with pdf [see 1 2 Azzalini and Chiogna (2004)] f GSTPSN (x; ξ, ω2 , λ1 , λ2 , ρ) λ1 (x − ξ ) λ2 |x − ξ | x −ξ c∗ (λ1 , λ2 , ρ) φ 2 , ,ρ , = ω ω ω ω

x ∈ R.

(28)

Based on a random sample x1 , . . . , xn from GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) in (28), we have the log-likelihood function to be n 1 log(2π ω2 ) − (xi − ξ )2 2 2ω2 i=1 n λ1 (xi − ξ ) λ2 |xi − ξ | , ,ρ , + log 2 ω ω

log L(θ) = n log c∗ (λ1 , λ2 , ρ) −

n

(29)

i=1

where θ = ξ, ω2 , λ1 , λ2 , ρ . In all the three examples discussed subsequently, the GSTPSN model has been fitted by the maximum likelihood method based on maximization of the log-likelihood function in (29).

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443

Table 1 MLEs of parameters for the heights of Australian female athletes data under GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) and two of its sub-models ξ

ω2

λ1

λ2

ρ

log(L( θ))

N (ξ, ω2 )

174.5939

67.2546

–

–

–

−352.3181

S N (ξ, ω2 , λ) GSTPSN ξ, ω2 , λ1 , λ2 , ρ

174.3918

67.2253

0.0314

–

–

−352.3180

165.9205

83.3820

0.4981

0.5387

−0.9646

−347.088

Fig. 2 Histogram of heights of 100 Australian female athletes. The lines represent distributions fitted by using maximum likelihood estimation: GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) (solid line); S N (ξ, ω2 , λ) (dashed line), which in this case is almost indistinguishable from N (ξ, ω2 )

Example 1 (Heights of Australian female athletes) We consider data on the heights (in centimeters) of 100 Australian female athletes. These data have been analyzed previously by Arellano-Valle et al. (2004), and are available at http://azzalini. stat.unipd.it/SN/index.html (and are also available in the R software in the sn package). Summary statistics of these data and some results on fits have been given by Arellano-Valle et al. (2004). Here, based on these data, we estimate the parameters by numerically maximizing the log-likelihood function in (29) with respect to the components of θ = ξ, ω2 , λ1 , λ2 , ρ . The results are summarized in Table 1 for GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) and two of its sub-models. From Table 1 as well as Fig. 2, we see that the model GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) fits the data much better than the two sub-models. Example 2 (Hemoglobin levels of Australian athletes) We consider data on the hemoglobin levels of 202 Australian athletes. These data have been analyzed earlier by Cook and Weisberg (1994). Here, based on these data, we estimate the parameters by numerically maximizing the log-likelihood function in (29) with respect to the components of θ = ξ, ω2 , λ1 , λ2 , ρ . The results are summarized in Table 2 for GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) and one of its submodel.

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Table 2 MLEs of parameters for the hemoglobin levels of Australian athletes data under GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) and one of its sub-model

S N (ξ, ω2 , λ) GSTPSN ξ, ω2 , λ1 , λ2 , ρ

ξ

ω2

λ1

λ2

ρ

log(L( θ))

13.6624

2.6641

0.9645

–

–

−348.2415

13.6623

2.3392

0.9042

2.1838

0.0040

−345.0192

Fig. 3 Histogram of hemoglobin levels of 202 Australian athletes. The lines represent distributions fitted by using maximum likelihood estimation: GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) (solid line); S N (ξ, ω2 , λ) (dashed line) Table 3 MLEs of parameters for the Otis IQ scores of non-white males data under GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) and two of its sub-models ξ

ω2

λ1

λ2

ρ

log(L( θ))

S N (ξ, ω2 , λ1 ) STPSN ξ, ω2 , λ1 , λ2

107.2482

68.0409

−0.0896

–

–

−183.3877

110.8128

71.9334

−0.5926

1.4826

–

−181.4620

GSTPSN (ξ, ω2 , λ1 , λ2 , ρ)

110.5548

51.3019

−0.3448

0.7990

−0.9272

−179.8876

From Table 2 as well as Fig. 3, we see that the model GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) fits the data much better than the sub-model S N (ξ, ω2 , λ). Example 3 (Otis IQ Scores of non-white males) Finally, we consider the data on the Otis IQ Scores of 52 non-white males hired by a large insurance company in 1971. These data, given by Roberts (1988) [see also Gupta and Brown (2001)], have been analyzed previously by Gupta and Gupta (2004) and Sharafi and Behboodian (2008). Here, based on these data, we estimate the parameters by numerically likelihood function in (29) with respect to maximizing the the components of θ = ξ, ω2 , λ1 , λ2 , ρ . The results are summarized in Table 3 for GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) and two of its sub-models.

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445

Fig. 4 Histogram of Otis IQ scores of 52 non-white males. The lines represent distributions fitted by using maximum likelihood estimation: GSTPSN (ξ, ω2 , λ1 , λ2 , ρ) (solid line); STPSN (ξ, ω2 , λ1 , λ2 ) (dashed line); S N (ξ, ω2 , λ1 ) (dotted line)

From Table 3 as well as Fig. 4, we see that the model GSTPSN ξ, ω2 , λ1 , λ2 , ρ fits the data much better than the two sub-models. Acknowledgements The authors express their sincere thanks to an anonymous referee for making some suggestions which led to an improvement in the presentation of this manuscript. The first two authors also acknowledge support from the Mahani Mathematical Research Center (Kerman, Iran) while the last author acknowledges support from the Natural Sciences and Engineering Research Council of Canada for conducting this research.

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