A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION BASED ON THE MULTIVARIATE SKEW NORMAL DISTRIBUTION

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1 J. Japan Statist. Soc. Vol. 45 No A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION BASED ON THE MULTIVARIATE SKEW NORMAL DISTRIBUTION Ahad Jamalizadeh* and Debasis Kundu** The Birnbaum-Saunders distribution has received some attention in the statistical literature since its inception. The univariate Birnbaum-Saunders distribution has been used quite effectively in analyzing positively skewed data. Recently,bivariate and multivariate Birnbaum-Saunders distributions have been introduced in the literature. In this paper we propose a new generalization of the multivariate (p-variate) Birnbaum-Saunders distribution based on the multivariate skew normal distribution. It is observed that the proposed distribution is more flexible than the multivariate Birnbaum-Saunders distribution,and the multivariate Birnbaum-Saunders distribution can be obtained as a special case of the proposed model. We obtain the marginal, reciprocal and conditional distributions,and also discuss some other properties. The proposed p-variate distribution has a total of 3p+ ( p 2) parameters. We use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters. One data analysis has been performed for illustrative purposes. Key words and phrases: Birnbaum-Saunders distribution,conditional probability density function,joint probability density function,maximum likelihood estimators, multivariate normal distribution,skew normal distribution. 1. Introduction Birnbaum and Saunders (1969a, 1969b) introduced a two-parameter lifetime distribution which has been used to analyze positively skewed data. The Birnbaum-Saunders (BS) distribution was derived through a monotone transform of the normal distribution. Since then a considerable amount of work has taken place on the development of the different aspects of this distribution, see for example Chang and Tang (1993, 1994), Dupis and Mills (1998), From and Li (2006), Ng et al. (2003, 2006), Leiva et al. (2008), Lemonte et al. (2007, 2008) and the references cited therein. A random variable T is said to have a two-parameter BS distribution with shape parameter α>0and scale parameter β > 0, if it has the cumulative distribution function (CDF) as follows: F T (t; α, β) =Φ(a(t; α, β)); t>0, Received August 19, Revised February 10, Accepted February 13, *Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman , Iran. **Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin , India. kundu@iitk.ac.in

2 2 AHAD JAMALIZADEH AND DEBASIS KUNDU where Φ( ) is the CDF of a standard normal distribution function and ( a(t; α, β) = 1 ) t β (1.1) α β. t Kundu et al. (2010) introduced a bivariate Birnbaum-Saunders (BBS) distribution by using the same monotone transformation. A bivariate random vector (T 1,T 2 ) T is said to have a BBS distribution, if the joint CDF can be written as follows; P (T 1 t 1,T 2 t 2 )=Φ 2 [a(t 1 ; α 1,β 1 ),a(t 2 ; α 2,β 2 ); ρ)]; t 1 > 0, t 2 > 0, where α 1 > 0, α 2 > 0, β 1 > 0, β 2 > 0, 1 <ρ<1, and Φ 2 (u, v; ρ) is the CDF of a standard normal random vector (Z 1,Z 2 ) T with correlation coefficient ρ. The authors discussed different properties of the BBS distribution and also addressed inferential issues. In a subsequent paper the authors, Kundu et al. (2013), proposed a multivariate Birnbaum-Saunders (MBS) model and discussed different properties. Several generalizations of the BS distribution have been proposed by different authors, see for example Diaz-Garcia and Leiva (2005), Leiva et al. (2008), Gomez et al. (2009) and Vilca et al. (2011). Vilca and Leiva (2006) introduced a new univariate BS distribution based on the skew normal distribution. The skew normal distribution has been proposed by Azzalini (1985). It is more flexible than the normal distribution, and the normal distribution can be obtained as a special case. Moreover, the skew normal distribution can have a heavier tail than the normal distribution. The proposed generalized multivariate Birnbaum-Saunders (GMBS) distribution is obtained by taking the same monotone transform as the BS distribution, and by replacing the multivariate normal distribution with the multivariate skew normal distribution. The random variable T is said to have a generalized Birnbaum-Saunders (GBS) distribution based on the skew normal distribution, if it has the PDF f T (t) =2φ(a(t; α, β))φ(λa(t; α, β))a(t; α, β); t>0. Here α>0, β>0, a(t; α, β) is the same as defined in (1.1), and { (β A(t; α, β) = d ) 1 1/2 ( ) } β 3/2 a(t; α, β) = + = t + β dt 2αβ t t 2α βt 3/2. It is observed that the GBS model is quite a flexible model, and the BS distribution can be obtained as a special case. Moreover, it can have a heavy tail depending on the parameter λ. Some recent developments on the GBS distribution can be obtained in Leiva et al. (2008) and Vilca et al. (2011). The aim of this paper is to introduce a multivariate Birnbaum-Saunders distribution based on the multivariate skew normal distribution using the same monotone transformation as the multivariate BS distribution and replacing the

3 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION 3 multivariate normal distribution with the multivariate skew normal distribution. The multivariate skew normal distribution was introduced by Azzalini and Dalla- Valle (1996), and it is a more flexible distribution than the multivariate normal distribution. Different properties of the generalized p-variate Birnbaum-Saunders (GMBS p ) distribution based on the multivariate skew normal (MSN) distribution have been established. It is observed that the multivariate BS distribution can be obtained as a special case of the proposed GMBS distribution. Marginal and conditional distributions are also provided. It is quite simple to generate samples from a GMBS p distribution, hence simulation experiments can be performed very easily. The proposed GMBS p model has 3p + ( p 2) unknown parameters. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, as expected. They can be obtained by solving 3p + ( p 2) non-linear equations simultaneously. We use the EM algorithm to compute the MLEs of the unknown parameters, which involves solving one p dimensional non-linear equation at each M stepof the EM algorithm. Therefore, the implementation of the EM algorithm becomes quite straight forward. The observed Fisher information matrix can be used to construct the asymptotic confidence intervals of the unknown parameters. Finally we address some testing of hypotheses issues also. We perform the analysis of one data set for illustrative purposes. The rest of the paper is organized as follows. In Section 2, we provide some preliminaries. GMBS p is introduced and different properties are discussed in Section 3. The use of the EM algorithm is provided in Section 4. The analysis of one data set has been presented in Section 5 and finally the paper concludes in Section Preliminaries 2.1. Multivariate BS distribution Let α, β R p, where α =(α 1,...,α p ) T and β =(β 1,...,β p ) T, with α i > 0, β i > 0, for i =1,...,p. Let Γ be a p p positive definite correlation matrix. The random vector T =(T 1,...,T p ) T is said to have a p-variate BS distribution with parameters (α, β, Γ), if it has the PDF (2.1) P (T t) =P (T 1 t 1,...,T p t p )=Φ p (a(t; α, β); Γ), where t =(t 1,...,t p ) T, t 1 > 0,...,t p > 0, and a(t; α, β) =(a(t 1 ; α 1,β 1 ),...,a(t p ; α p,β p )) T. Here u =(u 1,...,u p ) T and Φ p (u; Γ) denotes the joint CDF of a standard normal vector Z =(Z 1,...,Z p ) T, with mean zero and correlation matrix Γ. The joint PDF of T =(T 1,...,T p ) T can be obtained from (2.1) as (2.2) p f T (t; α, β, Γ) =φ p (a(t; α, β); Γ) A(t i ; α i,β i ), i=1

4 4 AHAD JAMALIZADEH AND DEBASIS KUNDU for t 1 > 0,...,t p > 0, and for u =(u 1,...,u p ) T φ p (u; Γ) = 1 (2π) p/2 Γ 1/2 e (1/2)u T Γ 1u, is the PDF of a standard normal vector with mean zero and correlation matrix Γ. From now on, the p-variate BS distribution with joint PDF (2.2) will be denoted by BS p (α, β, Γ). We will be further using the following notation φ p (x ; µ, Γ) = 1 (2π) p/2 Γ 1/2 e (1/2)(x µ)t Γ 1 (x µ) Multivariate skew normal distribution The multivariate skew normal distribution was introduced by Azzalini and Dalla Valle (1996). A p-dimensional random vector X =(X 1,...,X p ) T is said to have a multivariate skew normal (SN p ) distribution with parameter Γ, ap p positive definite correlation matrix, and λ =(λ 1,...,λ p ) T R p,ifx has the PDF (2.3) f SNp (x ; λ, Γ) =2φ p (x ; Γ)Φ(λ T x ); x R p. A multivariate skew normal distribution with PDF (2.3) will be denoted by SN p (Γ, λ). In the special case when λ = 0, the PDF (2.3) reduces to φ p (x, Γ), that is SN p (Γ, 0) =N p (0, Γ). Let us use the following notations. (2.4) X = ( X1 X 2 ), λ = ( λ1 λ 2 ), Γ = [ Γ11 Γ 12 Γ 21 Γ 22 Here the vectors X 1 and λ 1 are of the order q and the matrix Γ 11 is of the order p p. The rest of the quantities are defined so that they are compatible. The following lemma provides the marginal of X. ]. Lemma 1. X 1 SN q Γ 11, λ 1 + Γ 1 11 Γ 12λ 2 and 1+λ T 2 Γ 22.1 λ 2 X 2 SN p q Γ 22, λ 2 + Γ 1 22 Γ 21λ 1. 1+λ T 1 Γ 11.2 λ 1 Here Γ 22.1 = Γ 22 Γ 21 Γ 1 11 Γ 12 and Γ 11.2 = Γ 11 Γ 12 Γ 1 22 Γ 21. The following definition will be useful to provide the conditional distribution of X 2 given X 1 or vice versa. A p-dimensional random vector X =(X 1,...,X p ) T is said to have a multivariate extended skew normal distribution with parameters Γ R p p (Γ is a

5 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION 5 positive definite correlation matrix), λ =(λ 1,...,λ p ) T R p and τ R, denoted by ESN p (Γ, λ,τ), if its PDF is (2.5) f ESNp (x ; Γ, λ,τ)= φ p(x ; Γ)Φ(λ T x + τ) Φ(τ/ 1+λ T Γλ), x Rp. See for example Arnold and Beaver (2000). Lemma 2. Suppose X follows SN p (Γ, λ), and X, Γ, λ are partitioned as in (2.4). Then for x 1 R p, (a) [diag(γ 22.1 )] 1/2 (X 2 Γ 21 Γ 1 11 x 1) (X 1 = x 1 ) ESN p q ([diag(γ 22.1 )] 1/2 Γ 22.1 [diag(γ 22.1 )] 1/2, [diag(γ 22.1 )] 1/2 λ 2, (λ T 1 + λ T 2 Γ 21 Γ 1 11 )x 1). (b) The PDF of X 2 given X 1 = x 1, is f X2 (X 1 =x 1 )(x 2 )= φ p q(x 2 ; Γ 21 Γ 1 11 x1, Γ 22.1 )Φ(λ T 2 x 2 + λ T 1 x 1 ), Φ((λ T 1 + λ T 2 Γ 21 Γ 1 11 )x 1/ 1+λ T 2 Γ 22.1 λ 2 ) where φ p q ( ; Γ 21 Γ 1 11 x 1, Γ 22.1 ) is the PDF of N p q (Γ 21 Γ 1 11 x 1, Γ 22.1 ). The above results can be obtained directly, see Azzalini and Cap- Proof. itanio (1999). (2.6) Lemma 3. If X SN p (Γ, λ), then X d = Y + δh, where Y N p (0, Γ δδ T ), and H HN(0, 1), with δ = Γλ 1+λ T Γλ. Here HN(0, 1) denotes the half normal distribution with parameters 0 and 1 respectively, H = Z, where Z N(0, 1), and the PDF of H is as follows: (2.7) f H (h) = 2 π e h2 /2 ; h>0, see for example Azzalini and Dalla-Valle (1996). Lemma 4. If δ and λ are defined above, then there is a one to one correspondence between δ and λ, if Γ is non-singular. Proof. By simple algebraic calculations, it can be seen that Γλ δ = 1+λ T Γλ λ = Γ 1 δ 1 δ T Γ 1 δ, therefore, the result follows.

6 6 AHAD JAMALIZADEH AND DEBASIS KUNDU 3. Generalized multivariate BS distribution based on multivariate SN distribution In this section, we define the generalized multivariate BS distribution based on the multivariate SN distribution, and discuss its different properties Definition Definition 1. A p-variate random vector T = (T 1,...,T p ) T is said to have a generalized multivariate BS distribution based on the multivariate SN distribution with parameters α, β, Γ and λ if the CDF of T is (3.1) F T (t; α, β, Γ, λ) =F SNp (a(t; α, β); Γ, λ); t R p +. Here the parameters α, β, Γ, λ are the same as defined before and F SNp ( ; Γ, λ) denotes the CDF of SN p (Γ, λ). The PDF of T =(T 1,...,T p ) T becomes (3.2) p f T (t; α, β, Γ, λ) =f SNp (a(t; α, β); Γ, λ) A(t i ; α i,β i ); t R p +. i=1 From now it will be denoted by GMBS p (α, β, Γ, λ). It is immediately seen that when λ = 0, (3.2) coincides with the PDF of the multivariate BS distribution as defined by Kundu et al. (2013). Clearly, because of the presence of the parameter λ, it is more flexible than the multivariate BS distribution. In particular when p = 2, the PDF of T =(T 1,T 2 ) T, has the following form; ( ( ) ( ) ) 1 t1 β1 1 t2 β2 f T (t 1,t 2 )=2φ 2, ; ρ α 1 β 1 t 1 α 2 β 2 t 2 ( ( ) ( λ 1 t1 β1 Φ + λ )) 2 t2 β2 α 1 β 1 t 1 α 2 β 2 t 2 { 1 (β1 ) 1/2 ( ) } 3/2 β1 + 2α 1 β 1 t 1 t 1 { 1 (β2 ) 1/2 ( ) } 3/2 β2 +, 2α 2 β 2 t 2 t 2 where { } 1 φ 2 (u, v; ρ) = 2π 1 ρ exp 1 2 2(1 ρ 2 ) (u2 + v 2 2ρuv). We provide the surface plot of the joint PDF of GMBS 2 for different parameter values in Fig. 1. It is clear that it can take a variety of shapes, depending on the parameter values.

7 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION 7 (a) (b) (c) (d) (e) (f) Figure 1. The surface plot of BS-SN 2 for different parameter values when β 1 = β 2 = 1, and (a) α 1 =2=α 2, λ 1 = λ 2 =1,ρ =0.5, (b) α 1 =2=α 2, λ 1 = λ 2 =1,ρ =0.0, (c) α 1 =2=α 2, λ 1 = λ 2 =1,ρ = 0.5, (d) α 1 = α 2 =0.3, λ 1 = λ 2 =1,ρ =0.5, (e) α 1 = α 2 =0.3, λ 1 = 5.0, λ 2 =5.0, ρ =0.5, (f) α 1 = α 2 =0.3, λ 1 = 5.0, λ 2 =5.0, ρ = 0.5.

8 8 AHAD JAMALIZADEH AND DEBASIS KUNDU 3.2. Stochastic representation and simulation algorithm If T GMBS p (α, β, Γ, λ), then it has the following stochastic representation: ( T = d β1 4 [α 1X 1 + (α 1 X 1 ) 2 +4] 2,..., β ) T p 4 [α px p + (α p X p ) 2 +4] 2, where X =(X 1,...,X p ) T SN p (Γ, λ). Therefore, using Lemma 3, we immediately obtain; (3.3) T d = ( β1 4 [α 1(Y 1 + δ 1 H)+ (α 1 (Y 1 + δ 1 H) 2 +4] 2,..., β p 4 [α p(y p + δh)+ (α p (Y p + δh)) 2 +4] 2 ) T. Here δ =(δ 1,...,δ p ) T, Y =(Y 1,...,Y p ) T and H are the same as defined in Lemma 3. Therefore, the following steps can be adopted to generate T = (T 1,...,T p ) T from GMBS p (α, β, Γ, λ). Step 1. Make a Cholesky decomposition of Γ δδ T = AA T (say). Step 2. Generate p + 1 independent standard normal random variables say, U, U 1,...,U p. Step 3. Compute Y =(Y 1,...,Y P ) T = A(U 1,...,U p ) T. Step 4. Make the following transformation: (3.4) T i = β i 4 [α i(y i + δ i U )+ (α i (Y i + δ i U ) 2 +4] 2, for i =1,...,p. Then, T =(T 1,...,T p ) T has the required GMBS p (α, β, Γ, λ) distribution Marginal, conditional and reciprocals distributions In this section we provide the marginal and conditional distributions of the GMBS p (α, β, Γ, λ) distribution. Theorem 1. If T GMBS p (α, β, Γ, λ), and we let T, α, β, Γ, λ be partitioned as follows ( ) ( ) ( ) ( ) T1 α1 β1 λ1 (3.5) T =, α =, β =, λ =, T 2 α 2 β 2 λ 2 [ ] Γ11 Γ Γ = 12, Γ 21 Γ 22 where T 1, α 1, β 1, λ 1 are all q 1 vectors, Γ 11 is a q q matrixand the remaining elements are suitably defined. We have the following results.

9 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION 9 ) λ (a) T 1 GMBS q (α 1, β 1, Γ 11, 1 +Γ 1 11 Γ 12 λ 2 1+ λ T 2 Γ 22.1 λ 2 ) λ (b) T 2 GMBS p q (α 2, β 2, Γ 22, 2 +Γ 1 22 Γ 21 λ 1 1+ λ T 1 Γ 11.2 λ 1 (c) For t =(t 1,...,t p ) T =(t1 T, t 2 T )T R +p, where t 1 R +q and t 2 R +p q, and a(t : α, β) =(a1 T (t 1; α 1, β 1 ), a2 T (t 2; α 2, β 2 )) T, we have the conditional PDF of T 2 given T 1 = t 1, as f T2 T 1 =t 1 (t 2) = φp q(t2; Γ21Γ 1 p i=q+1 A(t i; α i,β i) 11 a 1(t 1; α 1, β 1 ), Γ 22.1)Φ(λ T 2 a 2(t 2; α 2, β 2 )+λ T 1 a 1(t 1; α 1, β 1 )) ( ) (λ T 1 + λ T 2 Γ 21Γ 1 11 Φ )a1(t1; α 1, β 1 ) 1+λ T 2 Γ 22.1λ 2 (d) The random variables T 1 and T 2 are independent if and only if Γ 12 = Γ 21 = 0, and λ 1 = 0 or λ 2 = 0. Proof. (a) It can be obtained by letting t q+1,...,t p, in (3.1) and using part (i) of Lemma 1. The proof (b) follows along the same line. To prove (c), observe that f T2 T 1 =t 1 (t 2 )= = f T (t; α, β, Γ, λ) f T1 (t 1 ; α 1, β 1, Γ 11, λ 1 ) f SNp (a(t; α, β); Γ, λ) f SNq (a 1 (t 1 ; α 1, β 1 ); Γ 11, λ 1 ) Now the result follows using Lemma 2. The proof of (d) follows from the result (c). p i=q+1 A(t i ; α i,β i ). Theorem 2. If T GMBS p (α, β, Γ, λ), and we let T, α, β, Γ, λ be partitioned as in (3.5), we can further use the following notation, if the vector a =(a 1,...,a p ) T, then a 1 =(a 1 1,...,a 1 p ) T, then we have the following results. ( ) { ( T1 β 1) [ ] ( )} (a) T2 1 1 Γ11 Γ GMBS p α,, 12 λ1,, β 2 Γ 21 Γ 22 λ ( 2 T 1) { ( 1 β 1) [ ] ( )} 1 Γ11 Γ (b) GMBS p α,, 12 λ1,, T 2 β 2 Γ 21 Γ 22 λ 2 (c) T 1 GMBS p (α, β 1, Γ, λ) Proof. (a) Let us denote ( ) ( Γ11 Γ Γ = 12 and Γ 1 A11 A = 12 Γ 21 Γ 22 A 21 A 22 We have, see Rao (1973), ( ) A11 Γ = Γ and Γ 1 A = 12. A 21 A 22 ).

10 10 AHAD JAMALIZADEH AND DEBASIS KUNDU Consider S q+1 = Tq+1 1,...,S p = Tp 1. We use the following notation; S 2 = (S q+1,...,s p ) T. To compute the joint PDF of (T1 T, S 2 T )=(T 1,...,T q,s q+1,..., S p ) first observe the following facts: (3.6) a(t 1 ; α, β) = a(t; α, β 1 ) and (3.7) φ p (u 1,...,u q, u q+1,..., u p ; Γ) =φ p (u 1,...,u q,u q+1,...,u p ; Γ). Therefore, the joint PDF of (T 1, S 2 ) is obtained from (3.2) as f (T1,S 2 )(t 1, s 2 ; α, β, Γ, λ) =f T (t 1, s2 1 ; α, β, Γ, λ) J. Since J = 1, using the PDF of T from (3.2) and the relations (3.6) and (3.7) the result follows. The proofs of (b) and (c) can be obtained along the same line. Theorem 3. If T GMBS p (α, β, Γ, λ), and H is the same as defined in (3.3), then the conditional PDF of T given H = h>0, is f T H=h (t; α, β, Γ, λ) =φ p (a(t; α, β); hδ, Γ δδ T p A(t i ; α i,β i ), i=1 for t =(t 1,...,t p ) T R p +. (3.8) Proof. From (3.3) it is immediate that ( {T H = h} = d β1 4 [α 1V 1 + (α 1 V 1 ) 2 +4] 2,..., β p 4 [α pv p + (α p V p ) 2 +4] 2 ) T, where V =(V 1,...,V p ) T N p (hδ, Γ δδ T ). Using the one to one correspondence between T and V, and using (2.1), it follows that (3.9) P (T t H = h) =P (T 1 t 1,...,T p t p H = h) =Φ p (a(t; α, β); δh, Γ δδ T ). Therefore, the result follows. Theorem 4. Let T GMBS p (α, β, Γ, λ), and H is the same as defined in (3.3). Let us define the random vector U =(U 1,...,U P ) T, where ( U 1 = 1 ) ( ) T 1 β1,...,u p = 1 T p β p. α 1 β 1 T 1 α p β p T p (a) The PDF of U is f U (u) =f SNp (u; λ, Γ) for u =(u 1,...,u p ) T R.

11 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION 11 (b) The conditional PDF of U, given H = h>0 is f U H=h (u; Γ, λ) =φ p (u; hδ, (Γ δδ T )) for u =(u 1,...,u p ) T R. Proof. (a) It can be obtained by using the transformation. (b) It immediately follows from Theorem 3. Theorem 5. If T GMBS p (α, β, Γ, λ), and H is the same as defined in (3.3), then the conditional PDF of H = h>0, given {T = t =(t 1,...,t p ) T } is {H T = t} = d U (U >0), where U N(δ T Γ 1 a(t; α, β), 1 δ T Γ 1 δ). Proof. Note that for t =(t 1,...,t p ) T R p + and h>0, f H T =t (h) = f T H=h(t)f H (h) f T (t) = 2 π φ p (a(t; α, β); hδ, Γ δδ T )e h2 /2. f SNp (a(t; α, β); Γ, δ) Using the following fact, see Rao (1973), (Γ δδ T ) 1 = Γ 1 + Γ 1 δδ T Γ 1 1 δ T Γ 1 δ, it can be seen after some simplification that { f H T =t (h) =K exp h 2 2(1 δ T Γ 1 δ) + h(a(t; α, β)t Γ 1 δ) (1 δ T Γ 1 δ) where K is independent of h. Now the result follows after completing the squares. If we use the following notations θ =(α, β, Γ, λ) and r(t) = φ(t) Φ(t), for t R, then using Theorem 5, the following can be easily obtained. }, (3.10) (3.11) E θ (H T = t) =δ T Γ 1 a(t; α, β) + (1 δ T Γ 1 δ)r δt Γ 1 a(t; α, β) (1 δ T Γ 1 δ) E θ (H 2 T = t) =(δ T Γ 1 a(t; α, β)) 2 +(1 δ T Γ 1 δ)+δ T Γ 1 a(t; α, β) (1 δ T Γ 1 δ)r δt Γ 1 a(t; α, β). (1 δ T Γ 1 δ) The conditional PDF of H given T = t, for t =(t 1,...,t p ) T R p +,is f H T =t (h) = φ(h; δt Γ 1 a(t; α, β), 1 δ T Γ 1 δ) Φ(λ T, h > 0. a(t; α, β))

12 12 AHAD JAMALIZADEH AND DEBASIS KUNDU 4. Inference 4.1. Estimation In this section we consider the estimation of the unknown parameters α, β, Γ and λ based on a random sample of size n, {t 1,...,t n }, from GMBS p (α, β, Γ, λ). We will be using the following notations; t T 1 =(t 11,...,t 1p ),...,t T n =(t np,...,t np ). The log-likelihood function of the observations without the additive constant becomes (4.1) l(α, β, Γ, λ) = n ln φ p (a(t i ; α, β); λ, Γ)+ i=1 + n i=1 j=1 p ln A(t ij ; α j,β j ). n ln Φ(λ T a(t i ; α, β)) The maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by maximizing the log-likelihood function (4.1) with respect to unknown parameters. It involves solving 3p + p(p 1)/2 non-linear equations. To avoid that we use the EM algorithm which involves maximizing a 2p dimensional optimization problem, at each step of the EM algorithm. The following observations will be useful to understand the basic idea of the EM algorithm. Since λ and δ have a one to one to correspondence, we are mainly restricted to estimate α, β, Γ and δ only for the EM algorithm. Let us assume that the complete data is as follows; i=1 (4.2) t (c) 1 =(t T 1,h 1 ) T,...,t (c) n =(t T n,h n ) T, where {t (c) (c) 1,...,t n } is a random sample of size n from (T,H), where T GMBS p (α, β, Γ, λ), and H is the same as defined in (3.3). We will show that based on the complete observations (4.2), the MLEs of α, β, Γ and δ can be obtained by solving a 2p dimensional optimization problem. The log-likelihood function of the complete data without the additive constant becomes (4.3) l c (α, β, Γ, δ) = n ln φ p (a(t i ; α, β); h i δ, Γ δδ T ) i=1 + n p ln A(t ij ; α j,β j ). i=1 j=1 We maximize the profile log-likelihood function to compute the MLEs of the unknown parameters, for the complete data set. First consider the following transformation of the data; u1 T =(u 11,...,u 1p ),...,un T =(u n1,...,u np ),

13 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION 13 where (4.4) ( ) u ij = 1 t ij β j ; i =1,...,n, j =1,...,p. α j β j t ij Now using Theorem 4, the log-likelihood function of the transformed data without the additive constant becomes (4.5) l ct (δ, Γ) = n ln φ p (u i ; h i δ, (Γ δδ T )). i=1 The MLEs of δ and Γ are as follows (4.6) δ(α, β) = n i=1 u ih i n i=1 h2 i and Γ(α, β) =S + δ(α, β) δ(α, β) T, where S = 1 n n (u i h i δ(α, β))(ui h i δ(α, β)) T. i=1 The MLEs of the unknown parameters can be obtained by maximizing the profile log-likelihood function of α and β, namely (4.7) l cp (α, β) = n ln φ p (a(t i ; α, β); h i δ(α, β), Γ(α, β) δ(α, β) δt (α, β)) i=1 + n p ln A(t ij ; α j,β j ). i=1 j=1 Suppose we denote the MLEs of α and β, which can be obtained by maximizing (4.7) as α and β, respectively, then the MLEs of Γ and δ become Γ = Γ( α, β) and δ = δ( α, β), respectively. Therefore, the MLEs of the unknown parameters can be obtained by solving a 2p dimensional optimization problem. Now we propose the following method to compute the MLEs of the unknown parameters of the GMBS p model. The method is mainly based on maximizing the profile log-likelihood function of α and β, where for given α and β, the MLEs of Γ and δ are performed using the EM algorithm. Algorithm. Assume some initial estimates of δ and Γ, sayδ (0) and Γ (0), re- Step 1. spectively. Step 2. Now obtain E(H T = t) and E(H 2 T = t) from (3.10) and (3.11), respectively, by replacing δ and Γ with δ (0) and Γ (0), respectively. Note

14 14 AHAD JAMALIZADEH AND DEBASIS KUNDU that the pseudo log-likelihood function of the transformed data obtained from (4.5) involves E(H T = t) and E(H 2 T = t). Step 3. Obtain δ (1) and Γ (1) from (4.6) by replacing h i and h 2 i T = t i ) and E(H 2 T = t i ). with E(H Step 4. Go back to Step1, and continue the process until converges, and obtain δ(α, β) and Γ(α, β). Step 5. Now maximize the profile log-likelihood function of α and β, l(α, β, Γ(α, β), δ(α, β)) as given in (4.3), to compute the MLEs of α and β. Now we discuss the asymptotic properties of the MLEs when all the parameters are unknown. Theorem 6. If θ =(α, β, Γ, λ) T is the parameter vector, and θ denotes the corresponding MLE, then d n( θ θ) Nm (0, I 1 ), d with m =3p + p(p 1)/2 being the dimension of the vector θ. Here, denotes the convergence in distribution while N m (0, I 1 ) denotes the m-variate normal distribution with mean vector 0, and the dispersion matrix I 1, with I being the Fisher information matrix. Proof. Since the GMBS p model satisfies all the regularity conditions for the MLEs to be consistent and asymptotically normally distributed, the result follows from the known asymptotic properties of the MLEs Testing of hypothesis In this subsection we discuss the likelihood ratio tests for some testing of hypotheses problems which will be of interest. We will be considering the following testing problem which might be useful in practice. Test I. H 0 : λ = 0 vs. H 1 : λ 0. This is an important testing problem, as it tests whether the data are coming from a multivariate Birnbaum-Saunders distribution or not. Since λ = 0 δ = 0, the MLEs of the unknown parameters can be obtained as follows. For a given β, the MLEs of α and Γ become (4.8) and α j (β) = 1 n ( ) 2 n t ij β j β j t ij i=1 1/2 ; j =1,...,p, (4.9) Γ(β) =P(β)Q(β)P T (β);

15 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION 15 here P(β) is a diagonal matrix given by P(β) = diag{1/ α 1 (β),..., α p (β)}, and the elements q jk (β) of the matrix Q(β) are given by ( )( tik ) q jk (β) = 1 n t ij β j β k (4.10) ; n β i=1 j t ij β k t ik for j, k =1,...,p. Finally the MLE of β can be obtained by maximizing the profile log-likelihood function of β, see Kundu et al. (2013) for details. If we denote α, β and Γ as the MLEs of α, β and Γ, respectively under H 0, then under H 0, for large n, (4.11) 2{l( α, β, Γ, 0) l( α, β, Γ, λ} χ 2 p. In Table 1 we present the critical values based on a 5% level of significance of the likelihood ratio test (4.11) for different parameter values. The critical values are obtained based on 1000 replications. We have taken p = 2, and we denote the matrix Γ =((γ ij )), for i, j =1, 2, where γ 11 = γ 22 = 1, and γ 12 = γ 21 = ρ. Table 1. Critical values of the test statistic (4.11) for different parameter values. n Set 1 Set 2 Set 3 Set 4 Set 5 Set Table 2. Size and power of the test for parameter Set 1 for different sample sizes. n λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 =1 =2 =3 =4 =5 = Table 3. Size and power of the test for parameter Set 2 for different sample sizes. n λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 =1 =2 =3 =4 =5 =

16 16 AHAD JAMALIZADEH AND DEBASIS KUNDU Table 4. Size and power of the test for parameter Set 3 for different sample sizes. n λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 λ 1 = λ 2 =1 =2 =3 =4 =5 = The value of the likelihood ratio test does not depend on the scale parameter. Hence, we take β 1 = β 2 = 1. We have considered six different parameter sets namely (i) Set 1: α 1 =1,α 2 =1,ρ =0.0, (ii) Set 2: α 1 =1,α 2 =1,ρ =0.5, (iii) Set 3: α 1 =1,α 2 =1,ρ =0.90, (iv) Set 4: α 1 =2,α 2 =2,ρ =0.0, (v) Set 5: α 1 =2,α 2 =2,ρ =0.5, (vi) Set 6: α 1 =2,α 2 =2,ρ =0.90. In Tables 2 to 4 we present the size and powers of the test H 0 : λ = 0 vs. H 1 : λ 0, for different parameter values. 5. Real data analysis In this section we present the analysis of a bivariate data set to see the effectiveness of the proposed model. The data set has been obtained from Johnson and Wichern (1999), and it represents two different measures of stiffness of 30 different boards. The first measurement involves sending a shock wave down the board, and the second measurement is determined while vibrating the board. The data set has been presented below in Table 5. Table 5. Two different measures of stiffness of 30 boards. T 1 T 2 T 1 T 2 T 1 T Table 6. Descriptive statistics of the data vector. Variable ME SD Q 2 Q 1 Q 3 T T

17 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION 17 Before progressing further, we compute the basic statistics of the data vector, and they are reported in Table 6. We present the mean (ME), standard deviation (SD), median (Q 2 ), first quartile (Q 1 ) and third quartile (Q 3 ) for both T 1 and T 2. Histograms of T 1 and T 2 are also provided in Fig. 2. From Q 1, Q 2 and Q 3, it is immediate that T 1 and T 2 are not symmetric. Both T 1 and T 2 are right skewed. The histograms of T 1 and T 2 also suggest that. We perform the test of symmetry for both of the marginals. We have used the distribution free test suggested by Randles et al. (1980). The test statistics for T 1 and T 2 are 1.71 and 1.77, and the associated p values are and , respectively. Therefore, it suggests that the marginals are not from symmetric distributions. The sample correlation coefficient between T 1 and T 2 is 0.932, which is very high. To get an idea about the shape of the empirical hazard function of the (a) (b) Figure 2. Histogram of (a) T 1 and (b) T 2. (a) (b) Figure 3. Scaled TTT plots of (a) T 1 and (b) T 2.

18 18 AHAD JAMALIZADEH AND DEBASIS KUNDU marginal, we provide the scaled TTT plots of T 1 and T 2 in Fig. 3. It indicates that both of them have increasing empirical hazard functions. We want to fit the proposed GMBS 2 distribution to the above data set. First we fit the bivariate Birnbaum-Saunders distribution to the above data set, and we obtain the estimates of the unknown parameters as follows: α 1 =3.7011, β1 = , α 2 =2.7314, β2 = , ρ = The associated log-likelihood value becomes To perform the EM algorithm, we have used the above values as the starting values of α 1, β 1, α 2, β 2 and ρ. Further we have set the staring values of λ 1 and λ 2 to be 0. The final estimates are as follows: α 1 =3.4420, β1 = , α 2 =3.0844, β2 = , ρ =0.9203, λ 1 = , λ2 = The associated log-likelihood value becomes intervals of α 1, β 1, α 2, β 2, ρ, λ 1 and λ 2 are The 95% confidence ( ), ( ), ( ), ( ), ( ), ( ), ( ), respectively. We perform the following test of the hypothesis Test. H 0 :(λ 1,λ 2 )=(0, 0) vs. H 1 :(λ 1,λ 2 ) (0, 0). Based on the likelihood ratio test as suggested in Subsection 4.2, the p value of the test statistic is less than 0.01, hence we reject the null hypothesis. The confidence intervals of λ 1 and λ 2 also suggest the same. It seems that the proposed GMBS 2 model provides a better fit than the bivariate BS distribution to the above stiffness data set. For comparison purposes we have also fitted (a) bivariate normal and (b) bivariate skew normal to this data set. We present the MLEs and the associated log-likelihood values in each case. Bivariate Normal. µ 1 = , µ 2 = , σ 1 = , σ 2 = , ρ =0.7886, log-likelihood = Bivariate SkewNormal. µ 1 = , µ 2 = , σ 1 = , σ 2 = , λ 1 = , λ 2 =3.8193, ρ =0.8943, log-likelihood = It is clear that based on the log-likelihood values, we prefer to use the GMBS 2 model to analyze this data set.

19 A MULTIVARIATE BIRNBAUM-SAUNDERS DISTRIBUTION Conclusions In this paper we have proposed a new multivariate distribution based on the multivariate skew normal and multivariate Birnbaum-Saunders distribution, and we name it as the generalized multivariate Birnbaum-Saunders distribution. The proposed distribution is more flexible than the multivariate Birnbaum-Saunders distribution, and the later can be obtained as a special case of the proposed distribution. We derive different properties of the proposed distribution, and use an EM algorithm to compute the MLEs of the unknown parameters. One data set has been analyzed, and it is observed that the proposed distribution provides a better fit than the multivariate Birnbaum-Saunders distribution. Acknowledgements The authors would like to thank the referees for their constructive suggestions which helped us to improve the earlier draft significantly. References Arnold,B. C. and Beaver,R. J. (2000). Hidden truncation models,sankhya Ser. A, 62, Azzalini,A. and Capitanio,A. (1999). Statistical applications of the multivariate skew normal distribution, J. Roy. Stat. Soc., Ser. B, 61, Azzalini,A. and Dalla Valle,A. (1996). The multivariate skew-normal distribution,biometrika, 83, Birnbaum,Z. W. and Saunders,S. C. (1969a). A new family of life distributions,j. Appl. Probab., 6, Birnbaum,Z. W. and Saunders,S. C. (1969b). Estimation for a family of life distributions with applications to fatigue, J. Appl. Probab., 6, Chang,D. S. and Tang,L. C. (1993). Reliability bounds and critical time for the Birnbaum- Saunders distribution, IEEE Trans. Reliab., 42, Chang,D. S. and Tang,L. C. (1994). Percentile bounds and tolerance limits for the Birnbaum- Saunders distribution, Commun. Stat. Theory and Methods, 23, Diaz-Garcia,J. A. and Leiva,V. (2005). A new family of life distributions based on the elliptically contoured distributions, J. Stat. Plann. Inference, 128, Dupuis,D. J. and Mills,J. E. (1998). Robust estimation of the Birnbaum-Saunders distribution, IEEE Trans. Reliab., 47, From,S. G. and Li,L. X. (2006). Estimation of the parameters of the Birnbaum-Saunders distribution, Commun. Stat. Theory and Methods, 35, Gomez,H. W.,Olivares,J. and Bolfarine,H. (2009),An extension of the generalized Birnbaum- Saunders distribution, Stat. Probab. Lett., 79, Johnson,R. A. and Wichern,D. W. (1999). Applied Multivariate Statistical Analysis,Fourth Edition,Prentice-Hall,New Jersey. Kundu,D.,Balakrishnan,N. and Jamalizadeh,A. (2010),Bivariate Birnbaum-Saunders distribution and associated inference, J. Multivar. Anal., 101, Kundu,D.,Balakrishnan,N. and Jamalizadeh,A. (2013),Multivariate Birnbaum-Saunders distribution: properties,inference and a generalization,j. Multivar. Anal., 116, Leiva,V.,Riquelme,M.,Balakrishnan,N. and Sanhueza,A. (2008). Lifetime analysis based on the generalized Birnbaum-Saunders distribution, Comput. Stat. Data Anal., 52, Lemonte,A. J.,Cribari-Neto,F. and Vasconcellos,K. L. P. (2007). Improved statistical inference for the two-parameter Birnbaum-Saunders distribution, Comput. Stat. Data Anal., 51,

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