Generalized Skew-Cauchy Distribution

Wen-Jang Huang∗,1 and Yan-Hau Chen2

1Department of Applied Mathematics, National University of Kaohsiung 2Institute of Statistics, National University of Kaohsiung

e-mail: [email protected], [email protected]

Abstract

Following the paper by Gupta et al. (2002) we investigate the generalized skew-symmetric distributions. Suppose Y is an absolutely continuous random variable with p.d.f. f and c.d.f. F , if a random variable X satisﬁes X2 =d Y 2, then X is said to have a generalized skew distribution of F (or f). For this work we consider the generalized skew-Cauchy (GSC) distribution and give some special examples of GSC distribution. Some of these examples are generated from generalized skew-normal or generalized skew-t distributions.

AMS 2000 subject classiﬁcations: Primary 60E05; secondary 62E10. Keywords: Skew-Cauchy distribution, skew-normal distribution, skew-symmetric distribution, skew-t distribution.

∗Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 94-2118-M-390-003 1. Introduction

The univariate skew-normal distribution has been studied by many authors, see e.g. Azzalini (1985,1986), Henze (1986), Chiogna (1998) and Gupta et al. (2004b). Following Azzalini (1985), a random variable X is said to have a skew-normal distribution with parameter λ, denoted by X ∼ SN (λ), if the probability density function (p.d.f.) is given by

fX (x) = 2φ(x)Φ(λx), λ, x ∈ R, (1) where φ and Φ are the p.d.f. and cumulative distribution function (c.d.f.) of the standard normal distribution, respectively. By letting λ x f (x) = 2φ(x)Φ(√ 1 ), λ , x ∈ , λ ≥ 0, (2) X 2 1 R 2 1 + λ2x Arellano-Valle et al. (2004) deﬁned a skew-generalized normal distribution, they denoted this distribution by SGN (λ1, λ2). The multivariate skew-normal distribution has also been considered by Azzalini and Dalla Valle (1996), Azzalini and Capitanio (1999), Gupta and Kollo (2003), and Gupta et al. (2004a). Here a p-dimensional random vector X is said to have a multivariate skew-normal distribution, denoted by X ∼ SN p(Ω, α), if it is continuous and its p.d.f. is given by

0 fX (x) = 2φp(x; Ω)Φ(α x), (3)

p where Ω > 0, α ∈ R , φp(x; Ω) is the p.d.f. of Np(0, Ω) distribution (the p-dimensional normal distribution with zero mean vector and correlation matrix Ω). Quadratic forms of skew-normal random vectors have been studied by Azzalini (1985), Azzalini and Dalla Valle (1996), Azzalini and Capitanio (1999), Loperﬁdo (2001), Genton et al. (2001), and Gupta and Huang (2002). Based on Gupta and Huang (2002), some parallel results for the class of multivariate skew normal-symmetric distributions have also been obtained by Huang and Chen (2006). If the p.d.f. of a random variable X has the form

fX (x) = 2f(x)G(x), x ∈ R, (4) where f is a p.d.f. of a random variable symmetric about 0, and G a Lebesque measurable function satisfying 0 ≤ G(x) ≤ 1 and G(x) + G(−x) = 1 a.e. on R, then X is said to have the so-called skew-symmetric distribution. Gupta et al. (2002) studied the models in which f is taken to be the p.d.f. from one of the following distributions: normal, Student’s t, Cauchy, Laplace, logistic, and uniform distribution, and G is a distribution function such that G0 is symmetric about 0. Nadarajah and Kotz (2003) considered the models that f is taken to be a normal p.d.f. with zero mean, while G is taken to come from one of the above continuous symmetric distributions. Multivariate skew-symmetric distributions have also been studied by Gupta and Chang (2003) and Wang et al. (2004a). The multivariate skew-Cauchy distribution

1 and multivariate skew t-distribution are studied by Arnold and Beaver (2000), and Gupta (2003), respectively.

It is known that the square of each of the N (0, 1), SN (λ) and SGN (λ1, λ2) distribution 2 is χ1 distributed. Based on this observation, in this paper, we introduce the generalized skew- symmetric model.

Deﬁnition 1. Suppose Y is an absolutely continuous symmetric random variable with p.d.f. f and c.d.f. F . Assume random variable X satisﬁes

X2 =d Y 2. (5)

Then X is said to have a generalized skew distribution of F (or f).

In Deﬁnition 1, if Y has a common distribution, such as normal distribution, then X will be said to have a generalized skew-normal distribution. The p.d.f. of a generalized skew distribution can be obtained by using the following lemma.

Lemma 1.(Huang et al. (2005)) Let n be a positive integer, and h(y), y ∈ A, a continuous p.d.f. Also assume A ⊂ [0, ∞), when n is even. Then Xn has h as its p.d.f., if and only if the p.d.f. of X is ( nxn−1h(xn) , n is odd, f (x) = (6) X n|x|n−1h(xn)G(x) , n is even,

n where x ∈ B = {x|x ∈ R, x ∈ A}, and G(x) is a Lebesgue measurable function which satisﬁes 0 ≤ G(x) ≤ 1 and G(x) + G(−x) = 1 a.e., ∀x ∈ B.

According to the above lemma, the following theorem is immediately.

Theorem 1. Assume the random variable Y is deﬁned as in Deﬁnition 1, and X has a generalized skew distribution of f. Then the p.d.f. of X is

fX (x) = 2f(x)G(x), x ∈ R, (7) where G is a Lebesque measurable function satisfying

0 ≤ G(x) ≤ 1 and G(x) + G(−x) = 1 a.e. on R. (8)

The p.d.f. given in (7) has the same form as (4). In fact, Arnold and Lin (2004) have used fX in (7) with f = φ to deﬁne the generalized skew-normal distribution. SN (λ), SGN (λ1, λ2), the skew normal-symmetric models of Nadarajah and Kotz (2003) all belong to the class of generalized skew-normal distribution. Gupta et al. (2002) gave three examples of skew-Cauchy distribution. In Section 2, we introduce the generalized skew-Cauchy (GSC) distribution. In Section 3, some examples of GSC

2 distribution will be given.

2. The GSC models

First we refer an equivalent condition to (5).

Proposition 1.(Wang et al. (2004b)) If X ∼ 2f(x)G(x) and Y ∼ 2f˜(x)G˜(x), where 2f(x)G(x) and 2f˜(x)G˜(x) are two p.d.f.s of generalized skew-symmetric distributions, then

f(x) = f˜(x) ⇔ τ(X) =d τ(Y ), for every even function τ, ⇔ X2 =d Y 2.

It should be mentioned here, one even function τ, such that τ(X) =d τ(Y ) is enough to imply X2 =d Y 2. The following example is the Proposition 3 of Arellano-Valle et al. (2004).

Example 1. Let X be a generalized skew-N (0, σ2) distribution, and Y a N (0, σ2) distribution. Then X2 =d Y 2, and |X| =d |Y |. Also |X| =d |Y | ∼ HN (0, σ2), where HN (0, σ2) denotes the half-normal distribution depend on the parameter σ.

We now use Deﬁnition 1 to deﬁne the generalized skew-Cauchy distribution. X is said to have a generalized skew-Cauchy distribution with parameter σ > 0, denoted by GSC(σ), if X2 =d Y 2, where Y has a C(0, σ) distribution. That is X2 has the p.d.f. σ g(t) = √ , t ≥ 0, σ > 0. (9) π[ t(σ2 + t)]

Denote the distribution of X2 by C2(0, σ). By Theorem 1, X has a GSC(σ) distribution, if and only if the p.d.f. of X has the following form 2σ f (x) = G(x), x ∈ , σ > 0, (10) X π(σ2 + x2) R where G(x), the skew function, is a Lebesque measurable function satisfying (8). There are some simple properties for the distribution of GSC(σ).

Proposition 2.

(i). The only symmetric GSC(σ) distribution is C(0, σ) distribution. 1 1 1 1 (ii). X ∼ GSC(σ) ⇔ X2 ∼ C2(0, σ) ⇔ ∼ C2(0, ) ⇔ ∼ GSC( ). X2 σ X σ

3 3. Examples of GSC distribution

Gupta et al. (2002) gave three examples of GSC distribution. The first example is defined in a similar way as the skew normal distribution defined by Azzalini (1985,1986). That is the p.d.f. of X is 2f(x)F (λx), where f(·) and F (·) are the p.d.f. and c.d.f. of C(0, σ) distribution, respectively. More precisely, the p.d.f. of X is given by

σ 2 arctan(λx/σ) f (x) = 1 + , λ, x ∈ , σ > 0. (11) 1 π(σ2 + x2) π R

As C(0, 1) distribution is exact the T1 distribution, inspired by this, the second example of GSC distribution is based on the skew-T1 distribution, the latter is deﬁned in a similar way as t distribution. √ Example 2. Let X = U/ W , where U has a generalized skew-N (0, 1) distribution and W 2 independent of U is χ1 distributed. Then X has a GSC(1) distribution. 2 d 2 √ Note that the random variable X given above satisﬁes X = X1 , where X1 = U1/ W1, 2 U1 has a N (0, 1) distribution, and W1 independent of U1 is χ1 distributed. That is X1 is T1 distributed. Hence GSC(1) is also a generalized skew-T1 distribution. For a special case, let U have a SN (λ) distribution. Then the p.d.f. of X is given by " # 1 λx f2(x) = 1 + , λ, x ∈ R, (12) π(1 + x2) p1 + (1 + λ2)x2 which is the second example of GSC distribution given by Gupta et al. (2002).

It can be seen easily that both p.d.f.s f1(x) and f2(x) in (11) and (12) can be rewritten as the same form of the p.d.f. given in (10). We give another example below.

Example 3. Let U and V be two independent random variables both distributed as a generalized skew-N(0, σ2) distribution. Then X = U/|V | has a GSC(1) distribution. The reason that the two X’s deﬁned in Example 2 and this example are equally distributed is due to the following proposition.

Proposition 3. Let W be χ2 distributed, and V be generalized skew-N (0, 1) distributed. Then √ 1 √ W and |V | are both half-normal distributed, i.e. W =d |V | ∼ HN (0, 1).

As an example, let U be SN (λ) distributed, and V generalized skew-N (0, 1) distributed. Then X = U/|V | has the p.d.f. given in (12).

Suppose that U and V are two independent random variables and both are N (0, σ2) distributed, σ > 0. It is known that not only U/|V | but also U/V are C(0, 1) distributed. Similar

4 result holds for generalized skew-normal distributions. The next example gives the distribution.

Example 4. Let U and V be two independent random variables distributed as generalized skew-N(0, σ2) distribution. Then X = U/V has a GSC(1) distribution.

The third way of Gupta et al. (2002) to deﬁne GSC distribution is by letting X = U/V , where U and V are independent random variables both distributed as SN (λ). Obviously X has a GSC(1) distribution. Although Gupta et al. (2002) failed to obtain the closed form of the p.d.f. of X, the p.d.f. nevertheless can be obtained. The following theorem indicates that the closed form of the p.d.f. of X, under a more general setting, can be derived.

Theorem 2. Let U and V be independent random variables distributed as SN (λ1) and SN (λ2), respectively, λ1, λ2 ∈ R. Then X = U/V has a GSC(1) distribution with p.d.f.

p 2 2 p 2 2 ! 1 2λ2 arctan(λ1x/ 1 + λ2 + x ) 2λ1x arctan(λ2/ 1 + (1 + λ1)x ) fX (x) = 1 + + , π(1 + x2) p 2 2 p 2 2 π 1 + λ2 + x π 1 + (1 + λ1)x λ1, λ2, x ∈ R. (13)

Again (13) takes the form (10). Before proving this theorem, we give some preliminary results below. The ﬁrst lemma can be found in Gupta and Brown (2001).

Lemma 2. For any b ∈ R, Z ∞ 1 1 φ(t)Φ(bt)dt = + arctan(b). (14) 0 4 2π

Lemma 3. For s ≥ 0, integer r ≥ 1 and a1, ··· , ar ∈ R, Z ∞ Γ((s + 1)/2)2(s−1)/2 vsφ(a v) ··· φ(a v)dv = . (15) 1 r r/2 Pr 2 (s+1)/2 0 (2π) ( i=1 ai ) √ −(r−1) Pr 2 1/2 Proof. Since φ(a1v) ··· φ(arv) = ( 2π) φ(( i=1 ai ) v), we only need to prove the case 2 2 r = 1. Now by letting t = a1v , we have Z ∞ Z ∞ s 1 s −a2v2/2 v φ(a1v)dv = √ v e 1 dv 0 2π 0 1 Z ∞ = √ t(s−1)/2e−t/2dt 2 (s+1)/2 2 2π(a1) 0 1 = √ Γ((s + 1)/2)2(s+1)/2 2 (s+1)/2 2 2π(a1) Γ((s + 1)/2)2(s−1)/2 = √ 2 (s+1)/2 2π(a1)

5 as desired.

Lemma 4. For s ≥ 2, integer r ≥ 1 and a1, ··· , ar, b ∈ R, Z ∞ s v φ(a1v) ··· φ(arv)Φ(bv)dv 0 bΓ(s/2)2s/2−1 s − 1 Z ∞ = + vs−2φ(a v) ··· φ(a v)Φ(bv)dv. (r+1)/2 Pr 2 Pr 2 2 s/2 Pr 2 1 r (2π) ( i=1 ai )( i=1 ai + b ) i=1 ai 0 Also Z ∞ 1 π b φ(a v) ··· φ(a v)Φ(bv)dv = + arctan , (16) 1 r (r+1)/2 Pr 2 1/2 Pr 2 1/2 0 (2π) ( i=1 ai ) 2 ( i=1 ai ) and Z ∞ 1 b vφ(a v) ··· φ(a v)Φ(bv)dv = 1 + . (17) 1 r r/2 Pr 2 Pr 2 2 1/2 0 2(2π) ( i=1 ai ) ( i=1 ai + b ) Proof. Again we only need to prove the case r = 1. For s ≥ 2, by integration by parts, it yields Z ∞ s v φ(a1v)Φ(bv)dv 0 Z ∞ s 1 −a2v2/2 = v √ e 1 Φ(bv)dv 0 2π a2v2 ∞ Z ∞ a2v2 −1 s−1 − 1 − 1 s−1 = √ v Φ(bv)e 2 − e 2 d v Φ(bv) 2 2πa1 0 0 Z ∞ Z ∞ b s−1 −a2v2/2 s − 1 s−2 −a2v2/2 = √ v e 1 φ(bv)dv + √ v e 1 Φ(bv)dv 2 2 2πa1 0 2πa1 0 Z ∞ Z ∞ b s−1 s − 1 s−2 = 2 v φ(a1v)φ(bv)dv + 2 v φ(a1v)Φ(bv)dv a1 0 a1 0 Γ(s/2)2s/2−1b s − 1 Z ∞ = + vs−2φ(a v)Φ(bv)dv. 2 2 2 s/2 2 1 2πa1(a1 + b ) a1 0

By letting t = a1v, from Lemma 2 we have Z ∞ 1 Z ∞ b 1 1 b φ(a1v)Φ(bv)dv = φ(t)Φ( t)dt = + arctan( ), 0 a1 0 a1 4a1 2πa1 a1 this is exact (16) for r = 1. Finally, again by integration by parts, Z ∞ vφ(a1v)Φ(bv)dv 0 ∞ Z ∞ −1 −a2v2/2 −a2v2/2 = √ Φ(bv)e 1 − e 1 dΦ(bv) 2 0 2πa1 0

6 Z ∞ −1 1 −a2v2/2 = √ − − b e 1 φ(bv)dv 2 2πa1 2 0 1 b Z ∞ = √ + φ(av)φ(bv)dv 2 2 2 2πa1 a1 0 1 b Γ(1/2)2−1/2 1 b = √ + = √ 1 + . 2 2 2 2 1/2 2 2 2 1/2 2 2πa1 a1 2π(a1 + b ) 2 2πa1 (a1 + b ) This completes the proof of this lemma.

We also have an extended corollary.

Corollary 1. For integer r ≥ 1, and a1, ··· , ar, b1, b2 ∈ R, Z ∞ vφ(a1v) ··· φ(arv)Φ(b1v)Φ(b2v)dv 0 " 1 b (π + 2 arctan(b /(Pr a2 + b2)1/2)) = π + 1 2 i=1 i 1 (r+2)/2 Pr 2 Pr 2 2 1/2 2(2π) ( i=1 ai ) ( i=1 ai + b1) # b (π + 2 arctan(b /(Pr a2 + b2)1/2)) + 2 1 i=1 i 2 . (18) Pr 2 2 1/2 ( i=1 ai + b2)

Proof. Again we only need to prove the case r = 1. By integration by parts, it yields Z ∞ vφ(a1v)Φ(b1v)Φ(b2v)dv 0 Z ∞ 1 −a2v2/2 = v √ e 1 Φ(b1v)Φ(b2v)dv 0 2π ∞ Z ∞ −1 −a2v2/2 −a2v2/2 = √ Φ(b1v)Φ(b2v)e 1 − e 1 d(Φ(b1v)Φ(b2v)) 2 0 2πa1 0 Z ∞ −1 1 −a2v2/2 = √ − − e 1 [b φ(b v)Φ(b v) + b φ(b v)Φ(b v)]dv 2 1 1 2 2 2 1 2πa1 4 0 1 b Z ∞ b Z ∞ = √ + 1 φ(a v)φ(b v)Φ(b v)dv + 2 φ(a v)φ(b v)Φ(b v)dv 2 2 1 1 2 2 1 2 1 4 2πa1 a1 0 a1 0 1 b 1 π b = √ + 1 + arctan 2 2 2 3/2 2 2 1/2 2 2 1/2 4 2πa1 a1 (2π) (a1 + b1) 2 (a1 + b1) b 1 π b + 2 + arctan 1 2 3/2 2 2 1/2 2 2 1/2 a1 (2π) (a1 + b2) 2 (a1 + b2) " # 1 b (π + 2 arctan(b /(a2 + b2)1/2)) b (π + 2 arctan(b /(a2 + b2)1/2)) = π + 1 2 1 1 + 2 1 1 2 . 3/2 2 2 2 1/2 2 2 1/2 2(2π) a1 (a1 + b1) (a1 + b2)

Proof of Theorem 2.

7 That X has a GSC(1) distribution is obvious. We derive the p.d.f. of X in the following. First the joint p.d.f. of U and V is

fU,V (u, v) = 4φ(u)Φ(λ1u)φ(v)Φ(λ2v), u, v ∈ R. Hence the p.d.f. of X is Z ∞ fX (x) = 4φ(xv)Φ(λ1xv)φ(v)Φ(λ2v)|v|dv. −∞ It follows Z ∞ Z ∞ fX (x) = 4φ(xv)Φ(λ1xv)φ(v)Φ(λ2v)vdv + 4φ(xv)Φ(−λ1xv)φ(v)Φ(−λ2v)vdv 0 0 Z ∞ = 4 vφ(xv)φ(v){Φ(λ1xv)Φ(λ2v) + [1 − Φ(λ1xv)][1 − Φ(λ2v)]}dv 0 Z ∞ = 4 vφ(xv)φ(v)[2Φ(λ1xv)Φ(λ2v) − Φ(λ1xv) − Φ(λ2v) + 1]dv 0 Z ∞ Z ∞ = 8 vφ(xv)φ(v)Φ(λ1xv)Φ(λ2v)dv − 4 vφ(xv)φ(v)Φ(λ1xv)dv 0 0 Z ∞ Z ∞ −4 vφ(xv)φ(v)Φ(λ2v)dv + 4 vφ(xv)φ(v)dv, x ∈ R. 0 0 By using Lemmas 3, 4 and Corollary 1, it yields

" p 2 2 1 λ2(π + 2 arctan(λ1x/ 1 + λ2 + x )) fX (x) = 8 · π + 8π2(1 + x2) p 2 2 1 + λ2 + x # " # λ x(π + 2 arctan(λ /p1 + (1 + λ2)x2)) 1 λ x + 1 2 1 − 4 · 1 + 1 p 2 2 4π(1 + x2) p 2 2 1 + (1 + λ1)x 1 + (1 + λ1)x " # 1 λ 1 −4 · 1 + 2 + 4 · 4π(1 + x2) p 2 2 2π(1 + x2) 1 + λ2 + x ! 1 2λ arctan(λ x/p1 + λ2 + x2) 2λ x arctan(λ /p1 + (1 + λ2)x2) = 1 + 2 1 2 + 1 2 1 , π(1 + x2) p 2 2 p 2 2 π 1 + λ2 + x π 1 + (1 + λ1)x λ1, λ2, x ∈ R, as desired.

We give another special case of Example 4.

Example 4.(Continued) Let X = U/V , where U is N (0, 1) distributed, V is SN (λ) distributed, and U and V are independent. By letting λ1 = 0 (or λ2 = 0) in (13), we obtain the p.d.f. of X 1 f (x) = , x ∈ . (19) X π(1 + x2) R

8 Consequently, X is C(0, 1) distributed and independent of λ. Actually by noting SN (0) =d N (0, 1), this is an immediate consequence of Theorem 2. Being C(0, 1) distributed, X and 1/X have the same distribution. Hence X1 = V/U is also C(0, 1) distributed.

We give a more general result below.

Example 5. Let U be N (0, σ2) distributed, and V be generalized skew-N (0, σ2) distributed. Then X = U/V is C(0, 1) distributed.

Proof. First the joint p.d.f. of U and V is 2 u v f (u, v) = φ( )φ( )G(v), u, v ∈ , σ > 0, U,V σ2 σ σ R where G(v) is a Lebesque measurable function satisfying 0 ≤ G(v) ≤ 1 and G(v) + G(−v) = 1 a.e. on R. Hence by letting t = v/σ, the p.d.f. of X is Z ∞ 2 xv v fX (x) = 2 φ( )φ( )G(v)|v|dv. −∞ σ σ σ Z ∞ = 2φ(xt)φ(t)G(σt)|t|dt −∞ Z ∞ Z ∞ = 2 tφ(xt)φ(t)G(σt)dt + 2 tφ(xt)φ(t)G(−σt)dt 0 0 Z ∞ = 2 tφ(xt)φ(t)[G(σt) + G(−σt)]dt 0 Z ∞ 1 = 2 tφ(xt)φ(t)dt = 2 , x ∈ R, 0 π(1 + x ) as desired.

Obviously the result still holds true if U is generalized skew-N (0, σ2) distributed and V is N (0, σ2) distributed.

We give the last proposition.

Proposition 4. Denote X by X(λ1, λ2) in Theorem 2. Then

d d lim X(λ1, λ2) = lim X(λ1, λ2) = C(0, 1). λ1→0 λ2→0

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