Multivariate Birnbaum-Saunders Distribution Based on Multivariate Skew Normal Distribution

Transcription

1 Multivariate Birnbaum-Saunders Distribution Based on Multivariate Skew Normal Distribution Ahad Jamalizadeh & Debasis Kundu Abstract Birnbaum-Saunders distribution has received some attention in the statistical literature since its inception. 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 total 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. Keywords: Birnbaum-Saunders distribution; joint probability density function; conditional probability density function; maximum likelihood estimators; skew normal distribution; multivariate normal distribution. Department of Statistics, Faculty of Mathematics & Computer, Shahid Bahonar University of Kerman, Kerman,Iran, Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin , India, kundu@iitk.ac.in. Corresponding author. 1

2 2 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 > 0 and scale parameter β > 0, if it has the cumulative distribution function CDF) as follows: F T t;α,β) = Φat;α,β)); t > 0, where Φ ) is the CDF of a standard normal distribution function and t at;α,β) = 1 ) β α β. 1) 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; PT 1 t 1,T 2 t 2 ) = Φ 2 [at 1 ;α 1,β 1 ),at 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

3 3 Vilca et al. 2011). Vilca and Leiva 2006) introduced a new univariate BS distribution based on skew normal distribution. The skew normal distribution has been proposed by Azzalini 1985). It is more flexible than the normal distribution, and normal distribution can be obtained as a special case. Moreover, the skew normal distribution can have 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, 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 skew normal distribution, if it has the PDF f T t) = 2φat;α,β))Φλat;α,β))At;α,β); t > 0. Here α > 0, β > 0, at;α,β) is same as defined in 1), and { β At;α,β) = d dt at;α,β) = 1 ) 1/2 ) } 3/2 β + 2αβ t t = t+β 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 development on 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 multivariate skew normal distribution using the same monotone transformation as the multivariate BS distribution with replacing the multivariate normal distribution with the multivariate skew normal distribution. Multivariate skew normal distribution was introduced by Azzalini and Dalla-Valle1996), and it is a more flexible distribution than the multivariate normal distribution. Different properties of the generalized p-variate Birnbaum-Saunders GMBS p ) distribu-

4 4 tion 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. ) p The proposed GMBS p model has 3p + unknown parameters. The maximum likelihood estimators MLEs) cannot be obtained in explicit forms, as expected. They can be 2 ) p obtained by solving 3p+ non-linear equations simultaneously. We use the EM algorithm 2 to compute the MLEs of the unknown parameters, which involves solving one p dimensional non-linear equation at each M step of 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 EM algorithm is provided in Section 4. The analysis of one data set has been presented in Section 5 and finally conclude the paper in Section 6. 2 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

5 5 has the PDF PT t) = PT 1 t 1,...,T p t p ) = Φ p at;α,β);γ), 2) where t = t 1,...,t p ) T, t 1 > 0,...,t p > 0, and at;α,β) = at 1 ;α 1,β 1 ),...,at 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) as f T t;α,β,γ) = φ p at;α,β);γ) for t 1 > 0,...,t p > 0, and for u = u 1,...,u p ) T p At i ;α i,β i ), 3) i=1 φ p u;γ) = 1 1 2π) p/2 Γ 1/2e 2 ut Γ 1 u, 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 3) will be denoted by BS p α,β,γ). We will be further using the following notation φ p x;µ,γ) = 1 1 2π) p/2 Γ 1/2e 2 x µ)t Γ 1 x µ). 2.2 Multivariate Skew Normal Distribution The multivariate skew normal distribution was introduced by Azzalini and Dalla Valle1996). 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, a p p positive definite correlation matrix, and λ = λ 1,...,λ p ) T R p, if X has the PDF f SNp x;λ,γ) = 2φ p x;γ)φλ T x); x R p. 4)

6 6 A multivariate skew normal distribution with PDF 4) will be denoted by SN p Γ,λ). In the special case when λ = 0, the PDF 4) reduces to φ p x,γ), that is SN p Γ,0) = N p 0,Γ). Let us use the following notations. X = X1 X 2 ), λ = λ1 λ 2 ), Γ = [ Γ11 Γ 12 Γ 21 Γ 22 ]. 5) Here the vectors X 1 and λ 1 are of the order q and the matrix Γ 11 is of the order p p. 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 11Γ 12 λ 2 and X 2 SN p q Γ 22, 1+λ T 2Γ 22.1 λ 2 Here Γ 22.1 = Γ 22 Γ 21 Γ 1 11Γ 12 and Γ 11.2 = Γ 11 Γ 12 Γ 1 22Γ 21. λ 2 +Γ 1 22Γ 21 λ 1. 1+λ T 1Γ 11.2 λ 1 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 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 f ESNp x;γ,λ,τ) = φ px;γ)φ λ T x+τ ) ), x R p, 6) Φ τ/ 1+λ T Γλ see for example Arnold and Beaver 2000). Lemma 2: Suppose X follows SN p Γ,λ), and X, Γ, λ are partitioned as in 5). Then for x 1 R p, ) a) [diagγ 22.1 )] 1 2 X2 Γ 21 Γ 1 11x 1 X1 = x 1 ) ESN p q [diagγ 22.1 )] 1 2 Γ22.1 [diagγ 22.1 )] 1 2,[diagΓ22.1 )] 1 2 λ2, ) ) λ T 1 +λ T 2Γ 21 Γ 1 x1. 11

7 7 b) The PDF of X 2 given X 1 = x 1, is f X2 X 1 =x 1) x 2) = φ ) ) p q x2 ;Γ 21 Γ 1 11x 1,Γ 22.1 Φ λ T 2x 2 +λ T 1x 1 λ ) T Φ 1 +λ T 2Γ 21 Γ 1 11 x1 / 1+λ T2Γ ), 22.1 λ 2 where φ p q ;Γ21 Γ 1 11x 1,Γ 22.1 ) is the PDF of Np q Γ21 Γ 1 11x 1,Γ 22.1 ). Proof: The above results can be obtained directly, see Azzalini and Capitanio 1999). Lemma 3: If X SN p Γ,λ), then X d = Y +δh, 7) where Y N p 0,Γ δδ T ), and H HN0,1), with δ = Γλ 1+λ T Γλ. Here HN0,1) denotes the half normal distribution with parameters 0 and 1 respectively, H = Z, where Z N0,1), and the PDF of H is as follows: 2 f H h) = /2 π e h2 ; h > 0, 8) 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 δ = Γλ 1+λ T Γλ λ = Γ 1 δ 1 δ T Γ 1 δ, therefore, the result follows.

8 3 Generalized Multivariate BS Distribution Based on Multivariate SN distribution 8 In this section, we define generalized multivariate BS distribution based on multivariate SN distribution, and discuss its different properties. 3.1 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 multivariate SN distribution with parameters α, β, Γ and λ if the CDF of T is F T t;α,β,γ,λ) = F SNp at;α,β);γ,λ); t R p +, 9) here the parameters α,β,γ,λ are 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 p f T t;α,β,γ,λ) = f SNp at;α,β);γ,λ) At i ;α i,β i ); t R p +. 10) From now it will be denoted by GMBS p α,β,γ,λ). It is immediate that when λ = 0, 10) 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. i=1 In particular when p = 2, the PDF of T = T 1,T 2 ) T, has the following form; ) 1 t1 β1 f T t 1,t 2 ) = 2φ 2, 1 ) ) t2 β2 ;ρ α 1 β 1 t 1 α 2 β 2 t 2 t1 ) λ 1 β1 Φ + λ t2 )) 2 β2 α 1 β 1 t 1 α 2 β 2 t 2 { 1 β1 ) 1/2 ) } { 3/2 β1 + 1 β2 ) 1/2 ) } 3/2 β2 +, 2α 1 β 1 t 1 t 1 2α 2 β 2 t 2 t 2

9 9 where { } 1 φ 2 u,v;ρ) = 2π 1 1 ρ 2exp 21 ρ 2 ) u2 +v 2 2ρuv). We provide the surface plot of the joint PDF of GMBS 2 for different parameter values in Figure 1. It is clear that it can take variety of shapes, depending on the parameter values. 3.2 Stochastic Representation and Simulation Algorithm If T GMBS p α,β,γ,λ), then it has the following stochastic representation: T d = β [ 1 α 1 X 1 + ] [ ) 2,..., 2 T β α 1 X 1 ) 4 2 p +4 α p X p + α p X p ) +4] 4 2, where X = X 1,...,X p ) T SN p Γ,λ). Therefore, using Lemma 3, we immediately obtain; T d = β [ 1 α 1 Y 1 +δ 1 H)+ ] [ ) 2,..., α 1 Y 1 +δ 1 H) 4 2 β 2 T p +4 α p Y p +δh)+ α p Y p +δh)) +4] 4 2, 11) Here δ = δ 1,...,δ p ) T, Y = Y 1,...,Y p ) T, H are same 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 = AU 1,...,U p ) T. Step 4: Make the following transformation: 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. 12) Then, T = T 1,...,T p ) T has the required GMBS p α,β,γ,λ) distribution.

10 Marginal, Conditional and Reciprocals Distributions In this section we provide the marginal and conditional distributions of GMBS p α,β,γ,λ) distribution. Theorem 1: If T GMBS p α,β,γ,λ), and let T,α,β,Γ,λ be partitioned as follows ) ) ) ) [ ] T1 α1 β1 λ1 Γ11 Γ T =, α =, β =, λ =, Γ = 12, 13) T 2 α 2 β 2 λ 2 Γ 21 Γ 22 where T 1,α 1,β 1,λ 1 are all q 1 vectors, Γ 11 is a q q matrix and the remaining elements are suitably defined. We have the following results. a) T 1 GMBS q α 1,β 1,Γ 11, λ ) 1+Γ 1 11Γ 12 λ 2 1+λ T 2Γ 22.1 λ 2 b) T 2 GMBS p q α 2,β 2,Γ 22, λ ) 2+Γ 1 22Γ 21 λ 1 1+λ T 1Γ 11.2 λ 1 c) For t = t 1,...,t p ) T = t T 1,t2) T T R + p, where t 1 R +q and t 2 R +p q, and at : α,β) = a T 1 t 1 ;α 1,β 1 ),a T 2 t 2 ;α 2,β 2 ) ) T, we have the conditional PDF of T2 given T 1 = t 1, as f T 2 T 1 =t 1 t 2 ) = φ ) p q t2 ;Γ 21 Γ 1 11a 1 t 1 ;α 1,β 1 ),Γ 22.1 Φ λ T 2a 2 t 2 ;α 2,β 2 )+λ T 1a 1 t 1 ;α 1,β 1 ) ) ) ) p i=q+1 At i ;α i,β i ) Φ λ T 1 +λ T 2 Γ 21 Γ λ T 2 Γ 22.1λ 2 a 1t 1 ;α 1,β 1 ) 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 9) and using part i) of Lemma 1. The proof b) follows along the same line. To prove c), observe that f T 2 T 1 =t 1 t 2 ) = f T t;α,β,γ,λ) f T 1 t 1 ;α 1,β 1,Γ 11,λ 1 ) = f SNp at;α,β);γ,λ) p f SNq a 1 t 1 ;α 1,β 1 );Γ 11,λ 1 ) i=q+1 At i ;α i,β i ).

11 11 Now the result follows using Lemma 2. Proof of d) follows from the result c). Theorem 2: If T GMBS p α,β,γ,λ), and let T,α,β,Γ,λ be partitioned as in 13). We furtherusethefollowingnotation. Ifthevectora = a 1,...,a p ) T,thena 1 = a 1 1,...,a 1 p ) T. We have the following results. a) T1 T 1 2 T 1 1 T 2 ) GMBS p { α, β 1 1 β 2 ), ) { ) β 1 1 b) GMBS p α,, β 2 c) T 1 GMBS p α,β 1,Γ, λ ) [ Γ11 Γ 12 Γ 21 Γ 22 [ Γ11 Γ 12 Γ 21 Γ 22 ] λ1, λ 2 ], λ1 λ 2 )}, )}, Proof: a) Let us denote ) Γ11 Γ Γ = 12 Γ 21 Γ 22 and Γ 1 A11 A = 12 A 21 A 22 ). We have, see Rao 1973), Γ = Γ and Γ 1 A11 A = 12 A 21 A 22 Consider S q+1 = T 1 q+1,...,s p = T 1 p. We use the following notation; S 2 = S q+1,...,s p ) T. To compute the joint PDF of T T 1,S T 2) = T 1,...,T q,s q+1,...,s p ) first observe the following facts: and ). at 1 ;α,β) = at;α,β 1 ) 14) φ p u 1,...,u q, u q+1,..., u p ;Γ) = φ p u 1,...,u q,u q+1,...,u p ; Γ). 15) Therefore, the joint PDF of T 1,S 2 ) obtained from 10) as f T 1,S 2) t 1,s 2 ;α,β,γ,λ) = f T t 1,s 1 2 ;α,β,γ,λ) J.

12 12 Since J = 1, using the PDF of T from 10) and the relations 14) and 15) 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 same as defined in 11), then the conditional PDF of T given H = h > 0, is f T H=h t;α,β,γ,λ) = φ ) p p at;α,β);hδ,γ δδ T At i ;α i,β i ), i=1 for t = t 1,...,t p ) T R p +. Proof: From 11) it is immediate that {T H = h} d = β [ 1 α 1 V 1 + ] [ ) 2,..., 2 T β α 1 V 1 ) 4 2 p +4 α p V p + α p V p ) +4] 4 2, 16) where V = V 1,...,V p ) T N p hδ,γ δδ T ). Using one to one correspondence between T and V, and using 2), it follows that PT t H = h) = PT 1 t 1,...,T p t p H = h) = Φ p at;α,β);δh,γ δδ T ). 17) Therefore, the result follows. Theorem 4: If T GMBS p α,β,γ,λ), and H is same as defined in 11). Let us define the random vector U = U 1,...,U P ) T, where a) The PDF of U is U 1 = 1 α 1 T 1 β 1 β1 T 1 ) ),...,U p = 1 T p β p α p β p T p. f U u) = f SNp u;λ,γ) for u = u 1,...,u p ) T R. 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.

13 13 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 same as defined in 11), 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 at;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) = f t)f T H=h Hh) 2 φ p at;α,β);hδ,γ δδ T )e h2 /2 H T=t =. f T t) π f SNp at;α,β);γ,δ) Using the fact, see Rao 1973), it can be seen after some simplification that f H T=t h) = Kexp { Γ δδ T ) 1 = Γ 1 + Γ 1 δδ T Γ 1 1 δ T Γ 1 δ, h 2 21 δ T Γ 1 δ) + hat;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 rt) = φt), for t R, then using Φt) Theorem 5, the following can be easily obtained. 1 δ E θ H T = t) = δ T Γ at;α,β)+ 1 T Γ 1 δ ) r δt Γ 1 at;α,β) 1 δ T Γ 1 δ ) 18) }, E θ H 2 T = t ) = δ T Γ 1 at;α,β) ) δ T Γ 1 δ ) +δ T Γ 1 at;α,β) 1 δ T Γ 1 δ ) r δt Γ 1 at;α,β) 1 δ T Γ 1 δ ). 19) 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;δ T Γ 1 at;α,β),1 δ T Γ 1 δ ) Φ λ T at;α,β) ), h > 0.

14 14 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 lα,β,γ,λ) = n n lnφ p at i ;α,β);λ,γ)+ lnφλ T at i ;α,β)) i=1 + n i=1 j=1 i=1 p lnat ij ;α j,β j ). 20) The maximum likelihood estimators MLEs) of the unknown parameters can be obtained by maximizing the log-likelihood function 20) with respect to unknown parameters. It involves solving 3p + pp 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 mainly restrict to estimate α, β, Γ and δ only for the EM algorithm. Let us assume that the complete data is as follows; t c) 1 = t T 1,h 1 ) T,...,t c) n = t T n,h n ) T, 21) where{t c) 1,...,t c) n }isarandomsampleofsizenfromt,h),wheret GMBS p α,β,γ,λ), and H is same as defined in 11). We will show that based on the complete observations 21), the MLEs of α, β, Γ and δ can be obtained by solving 2p dimensional optimization problem. The log-likelihood function of the complete data without the additive constant

15 15 becomes l c α,β,γ,δ) = n lnφ p at i ;α,β);h i δ,γ δδ T )+ i=1 n p lnat ij ;α j,β j ). 22) i=1 j=1 We maximize 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; u T 1 = u 11,...,u 1p ),...,u T n = u n1,...,u np ), where u ij = 1 t ij α j β j β j t ij ) ; i = 1,...,n, j = 1,...,p. 23) Now using Theorem 4, the log-likelihood function of the transformed data without the additive constant becomes l ct δ,γ) = The MLEs of δ and Γ are as follows δα,β) = n i=1 u ih i n i=1 h2 i n lnφ p u i ;h i δ,γ δδ T )). 24) i=1 and Γα,β) = S + δα,β) δα,β) T, 25) where S = 1 n n u i h i δα,β))ui h i δα,β)) T. i=1 The MLEs of the unknown parameters can be obtained by maximizing the profile loglikelihood function of α and β, namely l cp α,β) = n lnφ p at i ;α,β);h i δα,β), Γα,β) δα,β) δt α,β)) i=1 + n i=1 j=1 p lnat ij ;α j,β j ). 26) Suppose we denote the MLEs of α and β, which can be obtained by maximizing 26) are denoted by α, β, respectively, then the MLEs of Γ and δ become Γ = Γ α, β) and δ = δ α, β),

16 16 respectively. Therefore, the MLEs of the unknown parameters can be obtained by solving 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 EM algorithm. Algorithm Step 1: Assume some initial estimates of δ and Γ, say δ 0) and Γ 0), respectively. Step 2: Now obtain EH T = t) and EH 2 T = t) from 18) and 19), respectively, by replacing δ and Γ with δ 0) and Γ 0), respectively. Note that the pseudo log-likelihood function of the transformed data obtained from 24) involves EH T = t) and EH 2 T = t). Step 3: Obtain δ 1) and Γ 1) from 25) by replacing h i and h 2 i with EH T = t i ) and EH 2 T = t i ) Step 4: Go back to Step 1, and continue the process until converges, and obtain δα,β) and Γα,β). Step 5: Nowmaximizetheprofilelog-likelihoodfunctionofαandβ,lα,β, Γα,β),δα,β)) as given in 22), 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 n θ θ) d N m 0,I 1 ),

17 17 with m = 3p + pp 1)/2 being the dimension of the vector θ. Here, d 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: SinceGMBS p modelsatisfiesalltheregularityconditionsforthemlestobeconsistent and asymptotically normally distributed, the result follows from the known asymptotic properties of the MLEs. 4.2 Testing of Hypothesis In this subsection we discuss the likelihood ratio tests for some testing of hypotheses problems which will 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 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 and α j β) = 1 n n i=1 t ij β j β j t ij ) 2 1/2 ; j = 1,...,p, 27) Γβ) = Pβ)Qβ)P T β); 28) herepβ) isadiagonal matrixgiven bypβ) =diag{1/ α 1 β),..., α p β)}, andtheelements q jk β) of the matrix Qβ) are given by q jk β) = 1 n n i=1 t ij β j β j t ij ) tik β k βk t ik ) ; for j,k = 1,...,p. 29)

18 18 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, 2{l α, β, Γ,0) l α, β, Γ, λ} χ 2 p. 30) In Table 1 we present the critical values based on 5% level of significance of the likelihood ratio test 30) 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. The value of the likelihood ratio test does not dependonthescaleparameter. Hence, wetakeβ 1 = β 2 = 1. Wehaveconsideredsixdifferent 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, n Set 1 Set 2 Set 3 Set 4 Set 5 Set Table 1: Critical values of the test statistic 30) for different parameter values. 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

19 19 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 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. 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. 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 ), third quartile Q 3 ) for both T 1 and T 2. Histograms of T 1 and T 2 are also provided in Figure 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 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.

20 20 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 4: Size and power of the test for parameter Set 3 for different sample sizes. T 1 T 2 T 1 T 2 T 1 T Table 5: Two different measures of stiffness of 30 boards. 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 marginal, we provide the scaled TTT plots of T 1 and T 2 in Figure 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 = , β1 = , α 2 = , β2 = , ρ = The associated log-likelihood value becomes

21 21 Variable ME SD Q 2 Q 1 Q 3 T T Table 6: Descriptive statistics of the data vector. 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 used the staring values of λ 1 and λ 2 to be 0. The final estimates are as follows: α 1 = , β1 = , α 2 = , β2 = , ρ = , λ 1 = , λ2 = The associated log-likelihood value becomes The 95% confidence intervals of α 1, β 1, α 2, β 2, ρ, λ 1 and λ 2 are ), ), ), ), ), ), ), respectively. We perform the following testing of 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 Section 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 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.

22 22 Bivariate Normal: µ 1 = , µ 2 = , σ 1 = , σ 2 = , ρ = , log-likelihood = Bivariate skew normal: µ 1 = , µ 2 = , σ 1 = , σ 2 = , ρ = , log-likelihood = It is clear that based on the log-likelihood values, we prefer to use GMBS 2 model to analyze this data set. 6 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. 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 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. Acknowledgement The authors would like to thank the referees for their constructive suggestions which helped us to improve the earlier draft significantly.

23 23 References [1] Arnold, B. C. and Beaver, R. J.2000). Hidden truncation models, Sankhya Ser. A, 62, [2] Azzalini, A. and Capitanio, A. 1999). Statistical applications of the multivariate skew normal distribution, Jour. Royal. Stat. Soc., Ser. B, 61, [3] Azzalini, A. and Dalla Valle, A. 1996). The multivariate skew-normal distribution, Biometrika, 83, [4] Birnbaum, Z.W. and Saunders, S.C. 1969a). A new family of life distributions, Journal of Applied Probability, 6, [5] Birnbaum, Z.W. and Saunders, S.C. 1969b). Estimation for a family of life distributions with applications to fatigue, Journal of Applied Probability, 6, [6] Chang, D.S. and Tang, L.C. 1993). Reliability bounds and critical time for the Birnbaum-Saunders distribution, IEEE Transactions on Reliability, 42, [7] Chang, D.S. and Tang, L.C. 1994). Percentile bounds and tolerance limits for the Birnbaum-Saunders distribution, Communications in Statistics Theory and Methods, 23, [8] Diaz-Garcia, J.A. and Leiva, V. 2005). A new family of life distributions based on the elliptically contoured distributions, Journal of Statistical Planning and Inference, 128, [9] Dupuis, D.J. and Mills, J.E. 1998). Robust estimation of the Birnbaum-Saunders distribution, IEEE Transactions on Reliability, 47,

24 24 [10] From, S.G. and Li, L.X. 2006). Estimation of the parameters of the Birnbaum- Saunders distribution, Communications in Statistics Theory and Methods, 35, [11] Gomez, H.W., Olivares, J. and Bolfarine, H. 2009), An extension of the generalized Birnbaum-Saunders distribution, Statistics and Probability Letters, 79, [12] Johnson, R.A. and Wichern, D.W.1999). Applied Multivariate Statistical Analysis, Fourth Edition, Prentice-Hall, New Jersey. [13] Kundu, D., Balakrishnan, N. and Jamalizadeh, A. 2010), Bivariate Birnbaum- Saunders distribution and associated inference, Journal of Multivariate Analysis, 101, [14] Kundu, D., Balakrishnan, N. and Jamalizadeh, A.2013), Multivariate Birnbaum- Saunders distribution: properties, inference and a generalization, Journal of Multivariate Analysis, 116, [15] Leiva, V., Riquelme, M., Balakrishnan, N. and Sanhueza, A. 2008). Lifetime analysis based on the generalized Birnbaum-Saunders distribution, Computational Statistics & Data Analysis, 52, [16] Lemonte, A. J., Cribari-Neto, F. and Vasconcellos, K. L. P. 2007). Improved statistical inference for the two-parameter Birnbaum-Saunders distribution, Computational Statistics & Data Analysis, 51, [17] Lemonte, A. J., Simas, A. B. and Cribari-Neto, F. 2008). Bootstrap-based improved estimators for the two-parameter Birnbaum-Saunders distribution, Journal of Statistical Computation and Simulation, 78,

25 25 [18] Ng, H.K.T., Kundu, D. and Balakrishnan, N.2003). Modified moment estimation for the two-parameter Birnbaum-Saunders distribution, Computational Statistics & Data Analysis, 43, [19] Ng, H.K.T., Kundu, D. and Balakrishnan, N. 2006). Point and interval estimations for the two-parameter Birnbaum-Saunders distribution based on Type-II censored samples, Computational Statistics & Data Analysis, 50, [20] Randles, R.H., Flinger, M.A., Policello II, G.E. and Wolfe, D.A. 1980), An asymptotically distribution free test for symmetry versus asymmetry, Journal of the American Statistical Association, vol. 75, [21] Rao, C.R. 1973), Linear statistical inference and its applications, John Wiley & Sons, New York. [22] Vilca, F. and Leiva, V. 2006), A new fatigue life model based on family of skewelliptical distribution, Communications in Statistics - Theory and Methods, 35, [23] Vilca, F., Santana, L., Leiva, V. and Balakrishnan, N. 2011), Estimation of extreme percentile in Birnbaum-Saunders distributions, Computational Statistics and Data Analysis, 55,

26 t t t t a) t t2 c) t 1 e) t 2 t 1 b) d) t t 2 f) t2 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.

27 Relative frequency T 1 a) Relative frequency T 2 b) Figure 2: Histogram of a) T 1 and b) T a) b) Figure 3: Scaled TTT plots of a) T 1 and b) T 2.