A New Class of Multivariate Elliptically Contoured Distributions with Inconsistency Property
aa r X i v : . [ m a t h . S T ] A ug A New Class of Multivariate Elliptically ContouredDistributions with Inconsistency Property
Yeshunying Wang a and Chuancun Yin a a School of Statistics, Qufu Normal University
Shandong 273165, Chinae-mail: [email protected]
August 4, 2020
Abstract
We introduce a new class of multivariate elliptically symmetric distributionsincluding elliptically symmetric logistic distributions and Kotz type distributions.We investigate the various probabilistic properties including marginal distribu-tions, conditional distributions, linear transformations, characteristic functionsand dependence measure in the perspective of the inconsistency property. In ad-dition, we provide a real data example to show that the new distributions havereasonable flexibility.
Keywords:
Elliptically contoured distribution, Elliptically symmetric logistic distri-bution, Kotz type distribution, Inconsistency property, Generalized Hurwitz-Lerch zetafunction 1
Introduction
The multivariate normal distribution has been widely used in theory and practicebecause of its tractable statistical features. However, the light tail of the normaldistribution can not fit some practical situation well. The elliptically contoured dis-tributions (elliptical distributions), a new family of distributions with similar con-venient properties, overcomes the shortcomings of the normal distributions. An n -dimension random vector X is said to have a multivariate elliptical distribution, writ-ten as X ∼ Ell n ( µ , Σ , φ ) if its characteristic function can be expressed as ψ X ( t ) =exp( i t T µ ) φ ( t T Σ t ) , where µ is an n -dimension column vector, Σ is an n × n positivesemi-definite matrix, φ ( · ) is called characteristic generator. If X has a probabilitydensity function (pdf) f ( x ) , then f ( x ) = C n p | Σ | g n (cid:0) ( x − µ ) T Σ − ( x − µ ) (cid:1) , where C n is the normalizing constant and g n ( · ) is called density generator (d.g.). Thestochastic representation of X is given by X = µ + R A T U ( n ) , (1 . where A is a square matrix such that A T A = Σ , U ( n ) is uniformly distributed on theunit sphere surface in R n , R ≥ is independent of U ( n ) and has the pdf given by f R ( v ) = 1 R ∞ t n − g ( t ) dt v n − g ( v ) , v ≥ . (1 . Many members of the elliptical distributions such as the multivariate normal dis-tributions and student- t distributions, have been systematic studied. See the booksand papers of Cambanis et al. (1981), Fang et al. (1990), Kotz and Ostrovskii (1994),Liang and Bentler (1998), Nadarajah (2003). Nevertheless, research work on the mul-tivariate symmetric logistic distribution is far less than other members. The ellipticallysymmetric logistic distribution with density f ( x ) = Γ( n ) | Σ | − π n R ∞ u n exp( − u )(1+exp( − u )) du exp( − ( x − µ ) T Σ − ( x − µ ))[1 + exp( − ( x − µ ) T Σ − ( x − µ ))] , x ∈ R n , was introduced by Jensen (1985) and has been studied by Fang et al. (1990), Kano(1994), Yin and Sha (2018). Several applications of multivariate symmetric logisticdistribution in risk management, quantitative finance and actuarial science can befound in various literatures such as Landsman and Valdez (2003), Landsman et al.(2016a, 2016b, 2018).The paper will define a new class of elliptical distributions including the Kotz typedistributions and the logistic distributions, give the value of the normalizing constantand study the marginal distributions, conditional distributions, linear transformations,2haracteristic functions and local dependence functions in perspective of its inconsis-tency property.The rest of the paper is organized as follows. In Section 2, we introduce the defi-nition of a new class of multivariate elliptically symmetric distributions which includeelliptically symmetric logistic distributions and Kotz type distributions and discussthe expression of the normalizing constant. In Sections 3-7 we study the probabilisticproperties of the new class of elliptically distribution including marginal distributions,conditional distributions, linear transformations, characteristic functions in perspectiveof its inconsistency property. In addition, we give the expression of its local depen-dence function. In Section 8, we give the data analysis of the new class of ellipticallydistribution and we conclude in Section 9. We now give the definition of a new class of multivariate elliptically symmetric dis-tributions which include elliptically symmetric logistic distributions and Kotz typedistributions.
Definition 2.1
The n -dimensional random vector X is said to have a generalizedelliptical logistic (GL) distribution with parameter µ ( n -dimensional vector ) and Σ ( n × n matrix with Σ > ) if its pdf and density generator have the forms f ( x ) = C n | Σ | − g (cid:0) ( x − µ ) T Σ − ( x − µ ) (cid:1) , x ∈ R n , (2 . g ( t ) = t N − exp( − at s )(1 + exp( − bt s )) r , t > , (2 . respectively, where N + n > , a, b, s , s > , r ≥ are constants. Thenormalizing constant C n will be discussed in Section 2.2. Generalized logistic distribution (Yin and Sha (2018))Setting N = 1 , s = s = 1 in (2 . , we get g ( t ) = exp( − at )(1 + exp( − bt )) r , (2 . which is the density generator put forward by Yin and Sha (2018).2) Multivariate normal distribution
Setting N = 1 , a = , s = 1 , r = 0 in (2 . , we get g ( t ) = exp( − t ) , which is thedensity generator of the multivariate normal distribution.3) Multivariate exponential power (Epo) distribution (Landsman and Valdez(2003))For N = 1 , r = 0 , (2 . is the density generator of the multivariate exponentialpower distribution whose d.g. is usually written as g ( t ) = exp( − at s ) , a, s > . If s = and a = √ , we have the d.g. of the double exponential or Laplacedistribution defined as g ( t ) = exp( −√ t ) . Kotz type (Ko) distribution (Fang et al. (1990))For r = 0 , (2.2) is the density generator of the symmetric Kotz type distributionwhose d.g. is usually written as g ( t ) = t N − exp( − at s ) , a, s > , N + n > . (2 . When s = 1 , (2 . is the density generator of the original Kotz distribution whosed.g. is written as g ( t ) = t N − exp( − at ) , a > , N + n > . Elliptically symmetric logistic (Lo) distribution (Fang et al. (1990))Setting N = 1 , a = b = r = 1 , s = s = 1 , (2 . is the d.g. of the n -dimensionalelliptically symmetric logistic distribution , written as g n ( t ) = exp( − t )(1 + exp( − t )) . (2 . Generalized logistic type I (GLI) distribution (Arashi and Nadarajah (2016))Setting N = 1 , a = b = 1 , s = s = 1 in (2 . gives the density generator of thegeneralized logistic type I distribution written as g ( t ) = exp( − t )(1 + exp( − t )) r . (2 . Generalized logistic type III (GLIII) distribution (Arashi and Nadarajah(2016))Setting N = 1 , b = 1 , s = s = 1 , r = a in (2 . gives the density generator ofthe generalized logistic type III distribution written as g ( t ) = exp( − at )(1 + exp( − t )) a . (2 . Generalized logistic type IV (GLIV) distribution (Arashi and Nadarajah(2016))For N = 1 , s = s = 1 , r = p + a , where p > , (2.2) is the density generator of thegeneralized logistic type IV distribution written as g ( t ) = exp( − at )(1 + exp( − t )) p + a . (2 . s = s = s in (2 . , we obtain g n ( t ) = t N − exp( − at s )(1 + exp( − bt s )) r . (2 . For the sake of simplicity, we discuss probabilistic properties of GL distributionswith density generators defined as (2.9) in following sections.
To calculate the normalizing constant defined in (2 . , we introduce the Hurwitz-Lerchzeta function. The Hurwitz-Lerch zeta function and its integral representation arerespectively defined as Φ( z, s, a ) = ∞ X n =0 z n ( n + a ) s ( a ∈ C \ Z − , s ∈ C when | z | < ℜ ( s ) > when | z | = 1) , Φ( z, s, a ) = 1Γ( s ) Z ∞ t s − e − at − ze − t dt = 1Γ( s ) Z ∞ t s − e − ( a − t e t − z dt ( ℜ ( s ) > , ℜ ( a ) > when | z |≤ z = 1); ℜ ( s ) > when z = 1) . Various generalization and extensions of the Hurwitz-Lerch zeta function Φ( z, s, a ) havebeen studied by various researchers. The following expression of generalized Hurwitz-Lerch zeta function which will be used in this paper is defined by (cf. Lin et al. (2006)) Φ ∗ v ( z, s, a ) = 1Γ( v ) ∞ X n =0 Γ( v + n ) n ! z n ( n + a ) s ( v ∈ C , a ∈ C \ Z − , s ∈ C when | z | < ℜ ( s − v ) > when | z | = 1) , Φ ∗ v ( z, s, a ) = 1Γ( s ) Z ∞ t s − e − at (1 − ze − t ) v dt = 1Γ( s ) Z ∞ t s − e − ( a − v ) t ( e t − z ) v dt ( ℜ ( s ) > , ℜ ( a ) > when | z |≤ z = 1); ℜ ( s ) > when z = 1) . If v = 0 , Φ ∗ v ( z, s, a ) = Φ ∗ ( z, s, a ) = 1 a s , thus we denote Φ ∗ ( z, s, a ) by Φ ∗ ( s, a ) .Pointed out by Yin and Sha (2018), the normalizing constant of elliptical symmetriclogistic distribution suggested by Landsman and Valdez (2003), has no meaning when n =1 and n =2. Thus, we calculate the normalizing constant defined in (2.1) by thegeneralized Hurwitz-Lerch zeta function. The method was similarly used in Yin andSha (2018). Theorem 2.1
Letting X ∼ GL n ( µ , Σ , g n ) where g n is defined as (2 . , then thenormalizing constant defined in (2 . can be expressed as5 n = c ∗ n ( N, b, s ) (cid:20) Φ ∗ r ( − , s ( N + n − , ab ) (cid:21) − , where c ∗ n ( N, b, s ) = Γ( n )Γ( s ( N + n − b s ( N + n − sπ n , and Φ ∗ r is the generalized Hurwitz-Lerch zeta function. Proof.
Since Z ∞−∞ Z ∞−∞ · · · Z ∞−∞ f ( x ) d x = 1 , where f ( x ) is defined in (2 . and transformation from the rectangular to polar coor-dinates. We have C n = Γ( n ) π n (cid:20)Z ∞ x n − g n ( x ) dx (cid:21) − = Γ( n ) π n (cid:20)Z ∞ x N + n − e − ax s (1 + e − bx s ) r dx (cid:21) − = Γ( n ) π n b s ( N + n − s Γ( s ( N + n − (cid:20) Φ ∗ r (cid:18) − , s ( N + n − , ab (cid:19)(cid:21) − = c ∗ n ( N, b, s ) (cid:20) Φ ∗ r (cid:18) − , s ( N + n − , ab (cid:19)(cid:21) − , where c ∗ n ( N, b, s ) = Γ( n )Γ( s ( N + n − b s ( N + n − sπ n . Corollary 2.1.1
1) Supposing X ∼ Ko n ( µ , Σ , g n ) , where g n is defined as (2 . , then the normalizingconstant defined in (2 . can be expressed as C n = s Γ( n ) π n Γ( s ( N + n − (cid:20) Φ ∗ (cid:18) s ( N + n − , a (cid:19)(cid:21) − .
2) Supposing X ∼ Epo n ( µ , Σ , g n ) , where g n is g n ( t ) = exp( − at s ) , a, s > , then the normalizing constant defined in (2 . can be expressed as C n = s Γ( n ) π n Γ( n s ) (cid:20) Φ ∗ ( n s , a ) (cid:21) − .
3) Supposing X ∼ GLI n ( µ , Σ , g n ) , where g n is defined as (2 . , then the normalizingconstant defined in (2 . can be expressed as [ π n Φ ∗ r ( − , n , − .6) Supposing X ∼ GLIII n ( µ , Σ , g n ) , where g n is defined as (2 . , then thenormalizing constant defined in (2 . can be expressed as [ π n Φ ∗ a ( − , n , a )] − .5) Supposing X ∼ GLIV n ( µ , Σ , g n ) , where g n is defined as (2 . , then thenormalizing constant defined in (2 . can be expressed as [ π n Φ ∗ p + a ( − , n , a )] − .Consider a family of density generators { f ( u | p ) | p ∈ N } , (3 . where N denotes the set of all positive integers. According to Kano (1994) , we will saythat the family in (3 . possesses a consistency property if and only if Z ∞−∞ f p +1 X j =1 x j | p + 1 ! dx p +1 = f p X j =1 x j | p ! (3 . for any p ∈ N and almost all ( x , · · · , x p ) ∈ R p . We also say that the family is dimensioncoherent. However, the family of density generators defined in (2.2) does not satisfy7igure 1: a = b = r = 1 , N = 1 , s = s = 1 , ρ = 0 . .Figure 2: a = b = r = 1 , N = 2 , s = s = 1 , ρ = 0 . .Figure 3: a = b = 4 , r = 1 , N = 2 , s = s = 1 , ρ = 0 . .8he consistency property i.e. the marginal distributions of the n -dimension ( n > generalized elliptical bimodal logistic distribution — which are elliptically contoured— don’t have (2.2) as their density generators. In the following sections we will discussthe inconsistency property of the GL distribution and its applications. Supposing g ( x ) , the d.g. of GL distributed random variable X , is defined as follows g ( t ) = t N − exp( − at )(1 + exp( − bt )) r . Before we utilize (cf. Fang et al. (1990)) g m ( u ) = Z ∞ u ( ω − u ) n − m − g n ( ω ) dω, g m ( u ) = Z ∞ u g m +2 ( ω ) dω, (3 . to investigate g , g , · · · , it is sufficient to verify that g is a non-increasing functioni.e. g ′ ( u ) ≤ . Without loss of generality, assume a = b = 1 , then g ′ ( u ) = ( N − u N − e − u − t N − e − u + ( N − u N − e − u − t N − e − u + 2 ru N − e − u (1 + e − u ) r +1 .g ′ ( u ) ≤ if and only if u N − e − u (1 + e − u ) r +1 [( N − u + e − u ) − u (1 + e − u ) + 2 rue − u ] ≤ ,N ≤ u + e − u + u + ue − u − rue − u u + e − u . When u = 0 , N ≤ ; < u < , N ≤ ; u ≥ , N ≤ . Above all, g is a non-increasingfunction when N ≤ . Therefore, if N ≤ , t N − exp( − at )(1 + exp( − bt )) r = Z ∞ t g ( u ) du, g ( u ) = Z ∞ t ( u − t ) − g ( u ) du. Then, we have g ( u ) = − e − bu ) r +1 (cid:2) ( N − u N − e − au − au N − e − au + ( N − u N − e − ( a + b ) u − au N − e − ( a + b ) u + 2 bru N − e − ( a + b ) u (cid:3) , ( u ) = ae − au N − X j =0 ( N − j j ! Γ( j + ) b j + u N − − j Φ ∗ r +1 ( − e − bu , j + 12 , ab ) − ( N − e − au N − X k =0 ( N − k k ! Γ( k + ) b k + u N − − k Φ ∗ r +1 ( − e − bu , k + 12 , ab ) − ( N − e − ( a + b ) u N − X l =0 ( N − l l ! Γ( l + ) b l + u N − − l Φ ∗ r +1 ( − e − bu , l + 12 , a + bb )+ ae − au N − X v =0 ( N − v v ! Γ( v + ) b v + u N − − v Φ ∗ r +1 ( − e − bu , v + 12 , a + bb ) − bre − ( a + b ) u N − X q =0 ( N − q q ! Γ( q + ) b q + u N − − q Φ ∗ r +1 ( − e − bu , q + 12 , a + bb ) , where ( x ) n = x ( x − x − · · · ( x − n +1) . In the same way, we can obtain g , g , · · · , if g is a completely monotone function i.e. ( − n g ( n )1 ( t ) ≥ for every n ∈ N and t > ,where N denotes the set of non-negative integers. After that we can obtain g , g , · · · .If X has an elliptically symmetric logistic distribution i.e. X ∼ Lo n ( µ , Σ , g n ) , where g n is defined as (2.5), we can obtain g ( t ) = √ π (Φ ∗ ( − e − t , , − e − t √ ∗ ( − e − t , , ,g ( t ) = e − t − e − t (1 + e − t ) . In the same way, if X = ( X , · · · , X n ) T ∼ GL n ( µ , Σ , g n ) , the d.g. of X i ( i =1 , , · · · , n ) differs from g n obviously. We conclude the property in the followingtheorem. Theorem 3.1
Letting X = ( X T ( m ) , X T ( n − m ) ) T ∼ GL n ( µ , Σ , g n ) , where g n ( t ) is definedas (2 . , ≤ m < n , X ( m ) ∈ R m and X ( n − m ) ∈ R n − m , then the d.g. of X ( m ) is ˆ g m ( u ) = N − X j =0 ( N − j j ! u N − − j s Y k =0 Z ∞ y n − m + j − e − a ( s kk ! u s − k y k (1 + e − b P s l =0 ( s ll ! u s − l y l ) r dy, where ( x ) n = x ( x − x − · · · ( x − n + 1) and Φ ∗ r is the generalized Hurwitz-Lerchzeta function, Proof.
By formula (3 . , we have ˆ g m ( u ) = Z ∞ u ( t − u ) n − m − g n ( t ) dt = Z ∞ y n − m − ( y + u ) N − e − a ( y + u ) s (1 + e − b ( y + u ) s ) r dy = Z ∞ N − X j =0 ( N − N − · · · ( N − − j + 1) j ! u N − − j y n − m + j − e − a P s k =0 ( s kk ! u s − k y k (1 + e − b P s l =0 ( s ll ! u s − l y l ) r dy = N − X j =0 ( N − j j ! u N − − j s Y k =0 Z ∞ y n − m + j − e − a ( s kk ! u s − k y k (1 + e − b P s l =0 ( s ll ! u s − l y l ) r dy.
10n addition, setting s = s = 1 , ˆ g m ( u ) = N − X j =0 ( N − j j ! e − au u N − − j Z ∞ y n − m + j − e − ay (1 + e − bu e − by ) r dy = N − X j =0 ( N − j j ! u N − − j e − au Γ( n − m + j ) b n − m + j Φ ∗ r ( − e − bu , n − m j, ab )= N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j u N − − j e − au Φ ∗ r ( − e − bu , n − m j, ab ) follows.Setting s = s = s , r = 0 , ˆ g m ( u ) = N − X j =0 ( N − j j ! u N − − j Z ∞ y n − m + j − exp − s X k =0 a ( s ) k k ! y k u s − k ! dy = N − X j =0 ( N − j j ! u N − − j Z ∞ y n − m + j − exp − au s − a s X k =1 ( s ) k k ! y k u s − k ! dy ( letting x = u s − kk y )= N − X j =0 ( N − j j ! u N − − j exp( − au s ) s Y k =1 Z ∞ (cid:16) u − s − kk x (cid:17) n − m + j − exp (cid:18) − a ( s ) k k ! x k (cid:19) u − s − kk dy = N − X j =0 ( N − j j ! u N − − j exp( − au s ) s Y k =1 k u − s − kk ( n − m + j ) × Z ∞ (cid:18) k ! a ( s ) k (cid:19) k ( n − m + j ) − x k ( n − m + j ) exp( − x ) k ! r ( s ) k dx = N − X j =0 ( N − j j ! u N − − j exp( − au s ) s Y k =1 k u − s − kk ( n − m + j ) (cid:18) k ! a ( s ) k (cid:19) k ( n − m + j ) Γ( 1 k ( n − m j ))= N − X j =0 s Y k =1 ( N − j j ! (cid:18) k ! a ( s ) k (cid:19) k ( n − m + j ) Γ( k ( n − m + j )) k u N − − j − (1 − sk )( n − m + j ) exp( − au s ) follows.The theorem above tells us that g n ( · ) , ˆ g m ( · ) are different in form rather than di-mension changing. Corollary 3.1
1) Let X = ( X T ( m ) , X T ( n − m ) ) T ∼ Lo n ( µ , Σ , g n ) , where g n is defined as (2.5) then b g m ( u ) = Γ( n − m e − u Φ ∗ ( − e − u , n − m , .
11) Let X = ( X T ( m ) , X T ( n − m ) ) T ∼ Ko n ( µ , Σ , g n ) , where g n is defined as (2.4) then b g m ( u ) = N − X j =0 s Y k =1 ( N − j j ! (cid:18) k ! a ( s ) k (cid:19) k ( n − m + j ) Γ( k ( n − m + j )) k u N − − j − (1 − s k )( n − m + j ) exp( − au s ) . Proof.
We use Theorem 3.1 to conclude the density generator of X ( m ) .1) Setting N = a = b = r = 1 , s = s = 1 in Theorem 3.1 the result follows.2) Setting s = s = s , r = 0 , in Theorem 3.1 the result follows. Remark 3.1
According to Fang et al. (1990), for ≤ m ≤ n − , the marginaldensity generators are related by g m +2 ( x ) = − π g ′ m ( x ) , x > a.e. ) , (3 . where g ′ m ( · ) is the derivative of g m ( · ) . If N ≤ , (3.4) can be applied to the densitygenerators of GL distribution. Consider the partitions of X , µ , Σ as follows: X = (cid:18) X (1) X (2) (cid:19) , µ = (cid:18) µ (1) µ (2) (cid:19) , Σ = (cid:18) Σ Σ Σ Σ (cid:19) , (4 . where X (1) , µ (1) ∈ R m ( m < n ) , X (2) , µ (2) ∈ R n − m , Σ is an m × m matrix, Σ isan m × ( n − m ) matrix, Σ is an ( n − m ) × m matrix and Σ is an ( n − m ) × ( n − m ) matrix. Theorem 4.1
Let X ∼ GL n ( µ , Σ , g n ) where g n is defined as (2 . . Conditionally on X (2) = x (2) , we have the conditional distribution of X (1) .1) X (1) ∼ Ell m ( µ . , Σ . , g (1 . ) where µ . = µ (1) + Σ Σ − ( x (2) − µ (2) ) , Σ . = Σ − Σ Σ − Σ .
2) The density generator of X (1) | X (2) = x (2) can be written as g (1 . ( t ) = ( t + q ( x (2) )) N − e − a ( t + q ( x (2) ) ) s (cid:16) e − b ( t + q ( x (2) ) ) s (cid:17) r P N − j =0 ( N − j j ! q ( x (2) ) N − − j Q s k =0 I k,j , where q ( x (2) ) = ( x (2) − µ (2) ) ′ Σ − ( x (2) − µ (2) ) , I k,j , I k,j ( m, a, b, s , s , r ) = Z ∞ y m + j − e − a ( s kk ! q ( x (2) ) s − k y k (1 + e − b P s l =0 ( s ll ! q ( x (2) ) s − l y l ) r dy. roof.
1) The result follows by Fang et al. (1990).2) Define an n -dimension vector Y = (cid:18) Y (1) Y (2) (cid:19) = DX , D = (cid:18) I Σ Σ − I (cid:19) , where I is an m × m identity matrix, Σ , Σ are given in (4 . , to make Y (1) ∈ R m , Y (2) ∈ R n − m are independent with each other. We have Y (1) = X (1) − Σ Σ − X (2) , Y (2) = X (2) ; µ Y = D µ = (cid:18) I Σ Σ − I (cid:19) (cid:18) µ (1) µ (2) (cid:19) = (cid:18) ¯ µ (1 . µ (2) (cid:19) , ¯ µ (1 . = µ (1) − Σ Σ − µ (2) ; Σ Y = D Σ D ′ = (cid:18) I Σ Σ − I (cid:19) (cid:18) Σ Σ Σ Σ (cid:19) (cid:18) I − Σ Σ − I (cid:19) = (cid:18) Σ − Σ Σ − Σ Σ Σ (cid:19) (cid:18) I − Σ Σ − I (cid:19) = (cid:18) Σ . Σ (cid:19) , where Σ . = Σ − Σ Σ − Σ . Then Y ∼ GL n ( µ Y , Σ Y , g n ) , the d.g. of Y is same as X ’s d.g.. Moreover, the density generators of X (1) , X (2) are identicalwith density generators of Y (1) , Y (2) respectively, i.e. ˆ g X (1) ( t ) = ˆ g Y (1) ( t ) = ˆ g m ( t ) , ˜ g X (2) ( u ) = ˜ g Y (2) ( u ) = ˜ g n − m ( u ) . g n ( · ) , ˆ g m ( · ) , ˜ g n − m ( · ) are pairwise different in formrather than dimension changing. f Y = C n p | Σ Y | g n (cid:0) ( y − µ Y ) T Σ − Y ( y − µ Y ) (cid:1) = C n p | Σ Y | g n (cid:16) ( Y (1) − ¯ µ . ) T Σ − . ( Y (1) − ¯ µ . ) + ( Y (2) − µ (2) ) T Σ − ( Y (2) − µ (2) ) (cid:17) = C n p | Σ Y | g n (cid:16) ( X (1) − Σ Σ − X (2) − ¯ µ . ) T Σ − . ( X (1) − Σ Σ − X (2) − ¯ µ . )+( X (2) − µ (2) ) T Σ − ( X (2) − µ (2) ) (cid:17) = C n p | Σ Y | g n (cid:16) ( X (1) − µ . ) T Σ − . ( X (1) − µ . ) + ( X (2) − µ (2) ) T Σ − ( X (2) − µ (2) ) (cid:17) , µ . = µ (1) + Σ Σ − ( X (2) − µ (2) ) ,f Y (2) = ˜ C n − m p | Σ | ˜ g n − m (cid:16) ( X (2) − µ (2) ) T Σ − ( X (2) − µ (2) ) (cid:17) ,f Y (1) | Y (2) = ˆ C m p | Σ . | g (1 . (cid:16) ( X (1) − µ . ) T Σ − . ( X (1) − µ . ) (cid:17) . f Y (1) | Y (2) = f Y f Y (2) , we have g (1 . (cid:16) ( X (1) − µ . ) T Σ − . ( X (1) − µ . ) (cid:17) = g n (cid:16) ( X (1) − µ . ) T Σ − . ( X (1) − µ . ) + q ( x (2) ) (cid:17) ˜ g ( n − m ) ( q ( x (2) )) , then we have g (1 . ( t ) = ( t + q ( x (2) )) N − e − a ( t + q ( x (2) ) ) s (cid:16) e − b ( t + q ( x (2) ) ) s (cid:17) r P N − j =0 ( N − j j ! q ( x (2) ) N − − j Q s k =0 I k,j , and µ . = µ (1) + Σ Σ − ( x (2) − µ (2) ) , Σ . = Σ − Σ Σ − Σ . where q ( x (2) ) = ( x (2) − µ (2) ) T Σ − ( x (2) − µ (2) ) , I k,j , I k,j ( m, a, b, s , s , r ) = Z ∞ y m + j − e − a ( s kk ! q ( x (2) ) s − k y k (1 + e − b P s l =0 ( s ll ! q ( x (2) ) s − l y l ) r dy. In addition, setting s = s = 1 , g (1 . ( t ) = ( t + q ( x (2) )) N − e − at (1 + e − b ( t + q ( x (2) )) ) r P N − j =0 ( N − j j ! Γ( m )+ jb m j ( q ( x (2) )) N − − j Φ ∗ r ( − e − bq ( x (2) ) , m + j, ab ) holds, where q ( x (2) ) = ( x (2) − µ (2) ) T Σ − ( x (2) − µ (2) ) . Corollary 4.1
1) Supposing X ∼ Lo n ( µ , Σ , g n ) where g n is written as (2.5), the partitions of X , µ , Σ are same as (4 . . The density generator of X (1) conditionally on X (2) = x (2) is g (1 . ( t ) = h Γ( m ∗ ( − e − q ( x (2) ) , m , i − e − t (1 + e − ( t + q ( x (2) )) ) .
2) Supposing X ∼ Ko n ( µ , Σ , g n ) where g n is written as (2.4), the partitions of X , µ , Σ is same as (4 . . The density generator of X (1) conditionally on X (2) = x (2) is g (1 . ( t ) = (cid:0) t + q ( x (2) ) (cid:1) N − e − a ( t + q ( x (2) )) s P N − j =0 Q s k =1 γ k,j t N − − j − (1 − s k )( n − m + j ) e − at s , where γ x,y , γ x,y ( N, a, s , n, m, b ) = ( N − y y ! (cid:18) x ! a ( s ) x (cid:19) x ( n − m + y ) Γ( x ( n − m + y )) x . Characteristic functions and characteristic genera-tors
Theorem 5.1
Let X ∼ GL n ( µ , Σ , g n ) where g n is defined as (2 . . The characteristicfunction of X can be expressed as follows.1) If n = 1 , ψ X ( t ) = C e itµ Z ∞ x N − e − ax s (1 + e − bx s ) r cos( tσ √ x ) dx, t ∈ ( −∞ , ∞ ) .
2) If n > , ψ X ( t ) = e i t T µ ∞ X j =0 ( − j (2 j )! q j q Φ ∗ r ( − , s ( n + j + N − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( t T Σ t ) j , where Φ ∗ r is the generalized Hurwitz-Lerch zeta function, B ( · ) is the Beta function, t = ( t , t , · · · , t n ) T , t i ∈ ( −∞ , ∞ ) ,q x , q x ( N, n, b, s ) = Γ( s ( n + x + N − b s ( n + x + N − B ( n − , x + 12 ) . Proof. If n = 1 , ψ X ( t ) = E ( e itx ) = Z ∞−∞ e itx C σ g [( x − µσ ) ] dx = Z ∞−∞ C e itµ e itσx g ( x ) dx = C e itµ Z ∞−∞ [cos( tσx ) + i sin( tσx )] g ( x ) dx = C e itµ Z ∞ cos( tσ √ y ) y − g ( y ) dy = C e itµ Z ∞ cos( tσ √ y ) y N − e − ay s (1 + e − by s ) r dy. If n > , ψ X ( t ) = E ( e i t T X ) = e i t T µ E ( e i t T R Σ U ( n ) ) = e i t T µ E [Ω n ( R t T Σ t )]= e i t T µ Z ∞ Ω n ( v t T Σ t ) 1 R ∞ t n − g n ( t ) dt v n − v N − exp( − av s )(1 + exp( − bv s )) r dv = e i t T µ I , where Ω n ( k t k ) , t ∈ R n is the characteristic function of U ( n ) (Fang et al. (1990)) Ω n ( k t k ) = 1 B ( n − , ) Z π exp( i k t k cos θ ) sin n − θdθ, = Z ∞ Ω n ( v t T Σ t ) 1 R ∞ t n − g n ( t ) dt v n + N − exp( − av s )(1 + exp( − bv s )) r dv = Z ∞ B ( n − , ) (cid:20) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × Z π exp( iv ( t T Σ t ) cos θ ) sin n − θdθ v n + N − exp( − av s )(1 + exp( − bv s )) r dv = 1 B ( n − , ) (cid:20) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − Z π I sin n − θdθ,I = Z ∞ v n + N − exp( − av s + iv ( t T Σ t ) cos θ )(1 + exp( − bv s )) r dv = ∞ X l =0 i l ( t T Σ t ) l cos l θl ! Z ∞ v n + l + N − exp( − av s )(1 + exp( − bv s )) r dv = ∞ X l =0 i l ( t T Σ t ) l cos l θl ! Z ∞ s v s ( n + l + N − − exp( − av )(1 + exp( − bv )) r dv = ∞ X l =0 Γ( s ( n + l + N − b s ( n + l + N − s i l ( t T Σ t ) l cos l θl ! Φ ∗ r ( − , s ( n + l N − , ab ) . Then, we have I = 1 B ( n − , ) (cid:20) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × ∞ X l =0 Γ( s ( n + l + N − b s ( n + l + N − s i l ( t ′ Σ t ) l l ! Φ ∗ r ( − , s ( n + l N − , ab ) Z π sin n − θ cos l θdθ = 1 B ( n − , ) (cid:20) Γ( s ( n + N − b s ( n + N − Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × [ ∞ X j =0 Γ( s ( n +2 j +12 + N − b s ( n +2 j +12 + N − i j +1 ( t T Σ t ) j +12 (2 j + 1)! Φ ∗ r ( − , s ( n + 2 j + 12 + N − , ab ) · ∞ X j =0 Γ( s ( n +2 j + N − b s ( n +2 j + N − i j ( t T Σ t ) j (2 j )! Φ ∗ r ( − , s ( n + 2 j N − , ab ) B ( n − , j + 12 )]= (cid:20) B ( n − ,
12 ) Γ( s ( n + N − b s ( n + N − Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × ∞ X j =0 B ( n − , j + 12 ) Γ( s ( n +2 j + N − b s ( n +2 j + N − ( − j ( t T Σ t ) j (2 j )! Φ ∗ r ( − , s ( n + 2 j N − , ab ) . Finally, we obtain ψ X ( t ) = e it T µ ∞ X j =0 ( − j (2 j )! q j q Φ ∗ r ( − , s ( n + j + N − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( t T Σ t ) j , q x , q x ( N, n, b, s ) = Γ( s ( n + x + N − b s ( n + x + N − B ( n − , x + 12 ) . Corollary 5.1
Let X ∼ GL n ( µ , Σ , g n ) where g n is defined as (2 . . Thecharacteristic generator of X can be expressed as follows.1) If n = 1 , φ X ( t ) = C Z ∞ x N − e − ax s (1 + e − bx s ) r cos( tσ √ x ) dx, t ∈ ( −∞ , ∞ ) .
2) If n > , letting u n = ( u , u , · · · , u n ) T , φ X ( k u n k ) = ∞ X j =0 ( − j (2 j )! q j q Φ ∗ r ( − , s ( n + j + N − , ab )Φ ∗ r ( − , s ( n + N − , ab ) k u n k j , where q x , q x ( N, n, b, s ) = Γ( s ( n + x + N − b s ( n + x + N − B ( n − , x + 12 ) . Proof.
The results directly follow by the definition of characteristic generator.
Remark 5.1 Ω n ( k t k ) can be expressed in the following alternative forms: Ω n ( k t k ) = Γ( n ) √ π ∞ X k =0 ( − k k t k k (2 k )! Γ( k +12 )Γ( n +2 k ) , Ω n ( k t k ) = F ( n − k t k ) . We can obtain the following equivalent forms of the characteristic functions and char-acteristic generators with dimension n > . ψ X ( t ) = e i t T µ ∞ X k =0 ( − k (2 k )! γ k ( N, n, b, s ) Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( t T Σ t ) k , (5 . φ X ( k u n k ) = ∞ X k =0 ( − k (2 k )! γ k ( N, n, b, s ) Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( k u n k ) k , (5 . where γ k ( N, n, b, s ) = Γ( n ) π b s ( n + N − Γ( k + )Γ( s ( n + N + k − k + n )Γ( s ( n + N − .ψ X ( t ) = e i t T µ ∞ X k =0 Γ( s ( n + N + k − s ( n + N − b ks k ( n ) [ k ] k ! Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( t T Σ t ) k , (5 . X ( k u n k ) = ∞ X k =0 Γ( s ( n + N + k − s ( n + N − b ks k ( n ) [ k ] k ! Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( k u n k ) k . (5 . Similar as the density generator, the characteristic generator (c.g.) of the class ofmultivariate elliptically symmetric distribution defined in Definition 2.1 is not dimen-sionally coherent. In other word, the characteristic function ψ X ( · ) and the characteristicgenerator φ X ( · ) are related to the dimension of X . More details will be discussed inSection 7. Theorem 6.1
Let X ∼ GBL n ( µ , Σ , g n ) where g n is defined as (2 . .1) The expectation and the covariance are: E ( X ) = µ , Cov ( X ) = 1 n Γ( s ( N + n ))Φ ∗ r ( − , s ( N + n ) , ab ) b s Γ( s ( N + n − ∗ r ( − , s ( N + n − , ab ) Σ ;
2) For any integers m , · · · , m n , with m = P ni =1 m i , the product moments of Z := Σ − ( X − µ ) are E ( n Y i =1 Z m i i ) = 1 n Γ( s ( N + n + m − ∗ r ( − , s ( N + n + m − , ab )( n ) [ l ] b m s Γ( s ( N + n − ∗ r ( − , s ( N + n − , ab ) n Y i =1 (2 l i )!4 l i ( l i )! , where x [ n ] = x ( x + 1) · · · ( x + n − and Φ ∗ r is the generalized Hurwitz-Lerch zetafunction, if m i = 2 l i are even, i = 1 , · · · , n, m = 2 l ; E ( n Y i =1 Z m i i ) = 0 , if at least one of the m i is odd. Proof.
According to (1 . we have for real number p > , E ( R p ) = 1 R ∞ t n − g n ( t ) dt Z ∞ z n − p g n ( z ) dz = "Z ∞ t n − t N − e − at s (1 + e − bt s ) r dt − Z ∞ z N + n + p − e − az s (1 + e − bz s ) r dz ( setting x = bz s )= 2 b s ( N + n − s Γ( s ( N + n − (cid:20) Φ ∗ r ( − , s ( N + n − , ab ) (cid:21) − × Z ∞ s b − s (2 N + n + p − x s (2 N + n + p − − e − ab x (1 + e − x ) r dx = Γ( s ( N + n + p − ∗ r ( − , s ( N + n + p − , ab ) b p s Γ( s ( N + n − ∗ r ( − , s ( N + n − , ab ) .
18) Since X = µ + R A T U ( n ) and E ( U ( n ) ) = 0 , we have E ( X ) = µ + E ( R ) A T E ( U ( n ) ) = µ , and Cov ( X ) = Cov ( R A T U ( n ) ) = E ( R ) A T Cov ( U ( n ) ) A = 1 n E ( R ) Σ = 1 n Γ( s ( N + n ))Φ ∗ r ( − , s ( N + n ) , ab ) b s Γ( s ( N + n − ∗ r ( − , s ( N + n − , ab ) Σ .
2) By Eqs.(2.18) and (3.6) in Fang et al. (1990), E ( n Y i =1 Z m i i ) = E ( R m ) E ( n Y i =1 u m i i ) , where E ( Q ni =1 u m i i ) = n ) [ l ] Q ni =1 (2 l i )!4 li ( l i )! , if m i = 2 l i ( i = 1 , · · · , n ) are even, m = 2 l ; E ( Q ni =1 u m i i ) = 0 , if at least one of the m i is odd. Corollary 6.1
1) Let X ∼ Ko n ( µ, Σ , g n ) where g n is defined as (2 . .(1) The expectation and the covariance are: E ( X ) = µ , Cov ( X ) = 1 n Γ( s ( N + n )) a s Γ( s ( N + n − Σ ; (2) For any integers m , · · · , m n , with m = P ni =1 m i , the product moments of Z := Σ − ( X − µ ) are E ( n Y i =1 Z m i i ) = Γ( s ( N + n + m − n ) [ l ] a m s Γ( s ( N + n − n Y i =1 (2 l i )!4 l i ( l i )! , where x [ n ] = x ( x + 1) · · · ( x + n − , Φ ∗ r is the generalized Hurwitz-Lerch zetafunction, if m i = 2 l i ( i = 1 , · · · , n ) are even, m = 2 l ; E ( n Y i =1 Z m i i ) = 0 , if at least one of the m i is odd.2) Let X ∼ Lo n ( µ, Σ , g n ) where g n is defined as (2 . .(1) The expectation and the covariance are: E ( X ) = µ, Cov ( X ) = 1 n Γ(1 + n )Φ ∗ ( − , N + n , n )Φ ∗ ( − , n , Σ ; m , · · · , m n , with m = P ni =1 m i , the product moments of Z := Σ − ( X − µ ) are E ( n Y i =1 Z m i i ) = Γ( n + m )Φ ∗ ( − , n + m , n ) [ l ] Γ( n )Φ ∗ ( − , n , n Y i =1 (2 l i )!4 l i ( l i )! , if m i = 2 l i ( i = 1 , · · · , n ) are even, m = 2 l ; E ( Q ni =1 u m i i ) = 0 , if at least one ofthe m i is odd, where x [ n ] = x ( x + 1) · · · ( x + n − , Φ ∗ r is the generalizedHurwitz-Lerch zeta function. It has been mentioned in lots of researches that if X ∼ Ell n ( µ , Σ , φ ) , rank( Σ ) = k , B is an n × m matrix and v is an m × vector, then v + B T X ∼ Ell m ( v + B T µ , B T Σ B , φ ) . In the theorem above the characteristic generators of elliptical distributions are re-garded as unrelated to dimension i.e. the characteristic generator would not changewith the dimension during the liner transform. However the dimension coherent prop-erty is inapplicable for the characteristic generator of GL distribution. We will demon-strate GL distribution’s liner transform property in the following theorem.
Theorem 7.1
Assuming X ∼ GL n ( µ , Σ , φ ) with stochastic representation X = µ + R A T U ( n ) , Y = BX + b , where B is an m × n matrix, ≤ m < n ,rank ( B ) = m and b ∈ R n , the d.g. of X is g n ( x ) = x N − exp( − ax )(1 + exp( − bx )) r . Y ∼ Ell m ( B µ + b , B Σ B T , g ( m ) ,y ) , where g ( m ) ,y = N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j u N − − j e − au Φ ∗ r ( − e − bu , n − m , ab ) . Y ∼ Ell m ( B µ + b , B Σ B T , φ ( m ) ,y ) , where φ ( m ) ,y ( t ) = N − X j =0 ( N − j j ! Γ( n − + j ) b n − + j Φ ∗ r ( − e − bu , n −
12 + j, ab ) Z ∞ y N − j − e − ay cos( tσ Y √ y ) dy,t ∈ ( −∞ , ∞ ) , if m = 1 ; φ ( m ) ,y ( k ξ ( m ) k ) = P N − j =0 P ∞ l =0 P ∞ ω =0 ( − ω (2 ω )! α ∗ l β j q ∗ ω ( N, m, a, b, j, l ) P N − k =0 P ∞ j =0 α ∗ j β k q ∗ ( N, m, a, , j, l ) k ξ ( m ) k ω , ξ i ∈ ( −∞ , ∞ ) , ∗ x , α ∗ x ( N, n, m, a, b, r ) = ( − x Γ(2 r + x ) x !( x + ab ) n + N − , β x , β x ( N, n, m, b, r ) = ( N − x x ! Γ( n − m + x ) b n − m + x Γ(2 r ) ,q ∗ ω ( N, m, a, b, j, l ) = Γ( m + ω + N − j − B ( m − , ω +12 ) b m + N + ω − j − ( l + ab ) ω , ω = 0 , , , · · · , ξ ( m ) = ( ξ , ξ , · · · , ξ m ) ′ , if m > . Proof. Y is no longer GL distributed but elliptically distributed still.1) Applying Theorem 3.1 the result follows.2) When m =1, ψ Y ( t ) = E ( e it Y ) = Z ∞−∞ e ity ˜ C σ Y ˜ g [( y − µ Y σ Y ) ] dx = ˜ C e itµ Y Z ∞ cos( tσ Y √ y )˜ g ( y ) y − dy = ˜ C e itµ Y Z ∞ cos( tσ Y √ y ) N − X j =0 ( N − j j ! Γ( n − + j ) b n − + j y N − j − e − ay Φ ∗ r ( − e − by , n −
12 + j, ab ) dy = ˜ C e itµ Y N − X j =0 ( N − j j ! Γ( n − + j ) b n − + j Z ∞ y N − j − e − ay cos( tσ Y √ y )Φ ∗ r ( − e − by , n −
12 + j, ab ) dy, therefore, φ Y ( t ) = N − X j =0 C jN − Γ( n − + j ) b n − + j Φ ∗ r ( − e − bu , n −
12 + j, ab ) Z ∞ y N − j − e − ay cos( tσ Y √ y ) dy. When m > , ψ Y ( t ) = E ( e i t T Y ) = e i t T µ Y R ∞ t n − g n ( t ) dt × Z ∞ Ω n ( v t T Σ Y t ) v m − N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j v N − − j e − av Φ ∗ r ( − e − bv , n − m , ab ) dv = e i t T µ Y I − B ( m − , ) Z π sin m − θI dθ, I − = 1 R ∞ t m − g m ( t ) dt = "Z ∞ t m − N − X k =0 ( N − k k ! Γ( n − m + k ) b n − m + k t N − − k e − at Φ ∗ r ( − e − bt , n − m k, ab ) dt − = " N − X k =0 ( N − k k ! Γ( n − m + k ) b n − m + k r ) Z ∞ t N + m − k − e − at ∞ X j =0 Γ(2 r + j ) j ! ( − j e − bjt ( j + ab ) n − m + k dt − = " N − X k =0 β k ∞ X j =0 α j j + ab ) n − m + k Z ∞ t N + m − k − e − ( a + bj ) t dt − = " N − X k =0 ∞ X j =0 β k α j j + ab ) n − m + k a + bj ) m + N − − k Z ∞ t N + m − k − e − t dt − = " N − X k =0 ∞ X j =0 α j β k Γ( m + N − k − j + ab ) n + N − − = " N − X k =0 ∞ X j =0 α ∗ j β k Γ( m N − k − − ,β x , β x ( N, n, m, b, r ) = ( N − x x ! Γ( n − m + x ) b n − m + x Γ(2 r ) ,α x ( r ) , α x ( r ) = Γ(2 r + x )( − x x ! , α ∗ x , α ∗ x ( N, n, a, b, r ) = α x ( x + ab ) n + N − , = Z ∞ exp( iv ( t T Σ Y t ) cos θ ) × N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j v m + N − − j e − av Φ ∗ r ( − e − bv , n − m j, ab ) dv = N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j Z ∞ v N + m − j − e iv ( t T Σ t ) cos θ − av Φ ∗ r ( − e − bv , n − m j, ab ) dv = N − X j =0 β j Z ∞ v N + m − j − e iv ( t T Σ Y t ) cos θ − av ∞ X l =0 Γ(2 r + l ) l ! ( − l e − blv ( l + ab ) n − m + j dv = N − X j =0 ∞ X l =0 β j α l l + ab ) n − m + j Z ∞ v N + m − j − e − av − blv ∞ X q =0 i q v q q ! ( t T Σ Y t ) q cos q θdv = N − X j =0 ∞ X l =0 ∞ X q =0 β j α l ( t T Σ Y t ) q cos q θ ( l + ab ) n − m + j i q q ! Z ∞ v N + q + m − j − e − av − blv dv = N − X j =0 ∞ X l =0 ∞ X q =0 β j α l ( t T Σ Y t ) q cos q θ ( l + ab ) n − m + j i q q ! Γ( m + q + N − k − a + bl ) m + q + N − j − . Then we have Z π sin m − θI dθ = N − X j =0 ∞ X l =0 ∞ X q =0 α l β j ( t T Σ Y t ) q ( l + ab ) n − m + j i q q ! Γ( m + q + N − j − a + bl ) m + q + N − j − Z π cos q θ sin m − θdθ = N − X j =0 ∞ X l =0 ∞ X ω =0 ( − ω (2 ω )! α ∗ l β j Γ( m + ω + N − j − B ( m − , ω +12 ) b m + N + ω − j − ( l + ab ) ω ( t T Σ Y t ) ω ,ψ Y ( t ) = e i t T µ Y P N − j =0 P ∞ l =0 P ∞ ω =0 ( − ω (2 ω )! α ∗ l β j Γ( m + ω + N − j − B ( m − , ω +12 ) b m N + ω − j − ( l + ab ) ω P N − k =0 P ∞ j =0 α ∗ j β k Γ( m + N − k − B ( m − , ) ( t T Σ Y t ) ω = e i t T µ Y P N − j =0 P ∞ l =0 P ∞ ω =0 ( − ω (2 ω )! α ∗ l β j q ∗ ω ( N, m, a, b, j, l ) P N − k =0 P ∞ j =0 α ∗ j β k q ∗ ( N, m, a, , j, l ) ( t T Σ Y t ) ω , where q ∗ ω ( N, m, a, b, j, l ) = Γ( m + ω + N − j − B ( m − , ω +12 ) b m + N + ω − j − ( l + ab ) ω , ω = 0 , , , · · · . Therefore, φ ( m ) ,y ( k ξ ( m ) k ) = P N − j =0 P ∞ l =0 P ∞ ω =0 ( − ω (2 ω )! α ∗ l β j q ∗ ω ( N, m, a, b, j, l ) P N − k =0 P ∞ j =0 α ∗ j β k q ∗ ( N, m, a, b, j, l ) k ξ ( m ) k ω . emark 7.1 Similar as Remark 5.1, we can obtain the following equivalent forms ofthe characteristic functions and characteristic generators of Y = BX + b withdimension m > . ψ Y ( t ) = e i t T µ Y P ∞ l =0 P N − k =0 P ∞ p =0 ( − l (2 l )! α ∗ p β k A l Γ( m + N − k + l − P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j −
1) ( t T Σ Y t ) l , (7 . φ ( m ) ,y ( k ξ ( m ) k ) = P ∞ l =0 P N − k =0 P ∞ p =0 ( − l (2 l )! α ∗ p β k A l Γ( m + N − k + l − P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j − k ξ ( m ) k l ; (7 . ψ Y ( t ) = e i t T µ Y P ∞ l =0 P N − j =0 P ∞ k =0 α k β ∗ j Φ ∗ r ( − e − bv , n − m + j, ab ) P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j −
1) Γ( m ) π l ( m ) [ l ] ( t T Σ Y t ) l , (7 . φ ( m ) ,y ( k ξ ( m ) k ) = P ∞ l =0 P N − j =0 P ∞ k =0 α k β ∗ j Φ ∗ r ( − e − bv , n − m + j, ab ) P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j −
1) Γ( m ) π l ( m ) [ l ] k ξ ( m ) k l , (7 . where A x , A x ( N, m, a, b, p, k ) = B ( m , x + ) B ( m + x, ) b N − k + x ( p + ab ) x − m +1 ,β x , β x ( N, n, m, r ) = ( N − x x ! Γ( n − m + x )Γ(2 r ) b n − m + x , β ∗ x , β ∗ x ( N, n, m, b, r ) = β x Γ( N + m − x − b N + m − x − ,α x , α x ( r ) = ( − x Γ(2 r + x ) x ! , α ∗ x , α ∗ x ( N, m, a, b, r ) = α x ( x + ab ) n + N − . Here Φ ∗ r is the generalized Hurwitz-Lerch zeta function. Corollary 7.1 ( Marginal distributions ) Supposing X ∼ GL n ( µ , Σ , g n ) , where g n is defined as (2 . with s = s = 1 , the partitions of X , µ , Σ are given in (4.1), then X (1) ∼ Ell m ( µ (1) , Σ , b g m ) , X (2) ∼ Ell n − m ( µ (2) , Σ , b g n − m ) , where b g m ( u ) = N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j u N − − j e − au Φ ∗ r ( − e − bu , n − m j, ab ) , Φ ∗ r is the generalized Hurwitz-Lerch zeta function. Proof.
Taking B = ( I m , m × ( n − m ) ) , B = ( m × ( n − m ) , I n − m ) , X (1) = B X , X (2) = B X in Theorem 4.1 the result follows.On the basis of Theorem 5.1 with s = 1 and Theorem 7.1, it is clear that the c.g.of GL distributed random vector depends on dimension. Example ( Local dependence function ) Bairamov et al (2003) presented the localdependence function denoted by H ( x, y ) based on regression concepts as follows:24 ( x, y ) = E { ( X − E ( X | Y = y ))( Y − E ( Y | X = x )) } p E { ( X − E ( X | Y = y )) } p E { ( Y − E ( Y | X = x )) } . Alternative representations of H ( x, y ) are H ( x, y ) = Cov ( X, Y ) + ξ Y ( x ) ξ X ( y ) p V ar ( X ) + ξ X ( y ) p V ar ( Y ) + ξ Y ( x ) ,H ( x, y ) = ρ + φ X ( y ) φ Y ( x ) p φ X ( y ) p φ Y ( x ) , where ξ X ( y ) = E ( X | Y = y ) − E ( X ) , ξ Y ( x ) = E ( Y | X = x ) − E ( Y ) , ρ = Cov ( X, Y ) p V ar ( X ) p V ar ( Y ) ,φ X ( y ) = ξ X ( y ) p V ar ( X ) , φ Y ( x ) = ξ Y ( x ) p V ar ( Y ) . This function can characterize the dependence structure of two random variables X,Y localized at the fixed point. Suppose W = ( X, Y ) T ∼ GL ( µ , Σ , g ) , where g isdefined as (2 . with s = 1 , b I = R ∞ t b g ( t ) dt . Without loss of generality, let µ = (cid:18) (cid:19) , Σ = (cid:18) ρ ′ ρ ′ (cid:19) . We have
Cov ( W ) = N b Φ ∗ r ( − , N + 1 , ab )Φ ∗ r ( − , N, ab ) ρ ′ ,E ( Y | X = x ) = ξ Y ( x ) = ρ ′ x, E ( X | Y = y ) = ξ X ( y ) = ρ ′ y, respectively. V ar ( X ) = 2 C p − ρ ′ Z ∞ x b g ( x ) dx = 2 C b I = V ar ( Y ) .φ X ( y ) = ρ ′ y q C b I , φ Y ( x ) = ρ ′ x q C b I . I = Z ∞ t b g ( t ) dt = Z ∞ N − X j =0 ( N − j j ! Γ( + j ) b + j t N +1 − j ) e − at Φ ∗ r ( − e − bt , , ab ) dt = N − X j =0 ( N − j j ! Γ( + j ) b + j Z ∞ t N +1 − j ) e − at r ) ∞ X k =0 Γ(2 r + k )( − k k ! 1( k + ab ) + j e − bkt dt ( setting y = t )= N − X j =0 ∞ X k =0 β j α k k + ab ) + j Z ∞ y N − j − e − ay − bky dy = N − X j =0 ∞ X k =0 β j α k k + ab ) + j Γ( N − j + )( a + bk ) N − j + = N − X j =0 ∞ X k =0 β ∗ j α ∗ k , where α k = Γ(2 r + k )( − k k ! , α ∗ k = α k k + ab ) N +1 , β j = 1Γ(2 r ) ( N − j j ! Γ( + j ) b + j , β ∗ j = β j Γ( N − j + ) b N − j + . The local dependence function for the elliptically symmetric generalized logistic distri-bution can be expressed as follows, H ( x, y ) = N b Φ ∗ r ( − ,N +1 , ab )Φ ∗ r ( − ,N, ab ) ρ ′ + ρ ′ xy q C b I + ρ ′ y q C b I + ρ ′ y ,H ( x, y ) = N b Φ ∗ r ( − ,N +1 , ab )Φ ∗ r ( − ,N, ab ) C b I ρ ′ + ρ ′ xy C b I q ρ ′ y C b I q ρ ′ y C b I , where b I = ∞ X j =0 ∞ X k =0 β ∗ j α ∗ k , α ∗ k = ( − k Γ(2 r + k )2 k !( k + ab ) N +1 , β ∗ j = ( N − j j ! Γ( + j )Γ( N − j + )Γ(2 r ) b N +1 . We provide a numerical illustration for the GL distribution, using data in Table 1 whichis concluded by Gupta and Kundu (2010). It represents the strength measured in GPA,for single carbon fibers and impregnated 1000-carbon fiber tows. Table 1 shows thesingle fibers data set of 10 mm in gauge lengths with sample size 63.If a random variable X follows a general logistic distribution then its pdf defined as f ( x ; θ ) = e x/θ θ (1 + e x/θ ) , x ∈ R . Distribution Logistic NSL SL PRHL EEL GLParameter estimates b θ = 0 . b α = 1 . b α = 0 . b α = 3 . b α = 218 . b N = 1 . b β = 1 . b λ = 2 . b λ = 2 . b λ = 0 . b a = 1 . b λ = 1 . b µ = 2 . b µ = 2 . b θ = 0 . b s = 1 . b b = 8 . e + 04 b µ = 3 . b σ = 0 . b r = 4 . e − Log likelihood -165.5826 -123.4458 -58.0299 -58.9896 -56.8643 -49.6587AIC 333.1652 248.8916 122.0597 119.9792 119.7286 107.3174K-S 0.7123 0.5532 0.0918 0.0844 0.0735 0.0987K-S p-value 0.0000 0.0000 0.6632 0.7603 0.8853 0.5714
According to Chakraborty et al. (2012), the pdf of the new skew logistic (NSL) distri-bution is given by f SL ( x ; λ, α, β ) = [1 + sin( λx/ (2 β )) /α ] e − x/β β [1 + e − x/β ] , − ∞ < x < ∞ , α ≥ , λ ∈ R , β > . The GL distribution whose d.g. is defined as (2.9), is fitted to the data set and theresult is compared with those for the general logistic distribution, the NSL distribution,the skew logistic (SL) distribution, the proportional reversed hazard logistic (PRHL)distribution and the exponentiated-exponential logistic (EEL) distribution. The max-imum likelihood estimates, the log-likelihood value, the Akaike information criterion(AIC), the K-S test statistic and its p-value for the fitted distributions are presentedin Table 2. The results of the general logistic distribution, the NSL distribution, theSL distribution, the PRHL distribution and the EEL distribution are analyzed by In-dranil and Ayman (2018). Since the data set in Table 1 is widely used for generaland generalizations of logistic distributions, we consider the GL distribution with fixed N = 1 . , a = 1 . , s = 1 . . The results show that the GL distribution withfixed value of N , a and s fit data better among provided distributions in terms ofAkaike information criterion. However, as for the K-S test statistic, it doesn’t performwell as known distributions. 27 Concluding remarks
This paper defined the generalized logistic distribution whose density generator is de-fined as g ( t ) = t N − exp( − at s )(1 + exp( − bt s )) r , t > , (9 . where N + n > , a, b, s , s > , r ≥ are constants. By setting different a, b, s , s , r, N in (9 . , we obtained various density generators of elliptical distri-butions, such as the normal distribution, the Kotz type distribution, the exponentialpower distribution, the symmetric logistic distribution and generalized logistic typeI, III, IV distribution, etc. Our interest is to study the inconsistency properties andvarious probabilistic properties of this distribution including marginal distributions,conditional distributions, linear transformations, characteristic functions. In addition,we gave a data analysis which shows that the GL distributions are more flexible thanother distributions. We would give further research on statistic inference of this newkind of elliptical distributions in the subsequent research.28 ppendix A. Proofs Appendix A.1. Proof of (5 . − (5 . Proof.
When the dimension n > , Ω n ( k t k ) defined as Ω n ( k t k ) = Γ( n ) √ π ∞ X k =0 ( − k k t k k (2 k )! Γ( k +12 )Γ( n +2 k ) , we have the characteristic function as follows. ψ X ( t ) = E ( e i t T X ) = e i t T µ E ( e i t T R Σ U ( n ) ) = e i t T µ E [Ω n ( R t T Σ t )]= e i t T µ Z ∞ Ω n ( v t T Σ t ) 1 R ∞ t n − g n ( t ) dt v n − v N − exp( − av s )(1 + exp( − bv s )) r dv = e i t T µ (cid:20)Z ∞ t n + N − exp( − at s )(1 + exp( − bt s )) r (cid:21) − Z ∞ Ω n ( v t T Σ t ) v n + N − exp( − av s )(1 + exp( − bv s )) r dv = e i t T µ (cid:20) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × Z ∞ Γ( n ) π ∞ X k =0 ( − k ( t T Σ t ) k v k Γ( k +12 )(2 k )!Γ( n +2 k ) v n + N − exp( − av s )(1 + exp( − bv s )) r dv = e i t T µ (cid:20) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × ∞ X k =0 Γ( n ) π ( − k ( t T Σ t ) k Γ( k +12 )(2 k )!Γ( n +2 k ) Z ∞ v n + N + k − exp( − av s )(1 + exp( − bv s )) r dv = e it T µ (cid:20) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × ∞ X k =0 Γ( n ) π ( − k ( t T Σ t ) k Γ( k +12 )(2 k )!Γ( n +2 k ) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab )= e i t T µ Γ( n ) π ∞ X k =0 ( − k (2 k )! Γ( k + )Γ( s ( n + N + k − k + n )Γ( s ( n + N − b s ( n + N − × Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( t T Σ t ) k = e i t T µ ∞ X k =0 ( − k (2 k )! γ k Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( t T Σ t ) k , where γ k ( N, n, b, s ) = Γ( n ) π b s ( n + N − Γ( k + )Γ( s ( n + N + k − k + n )Γ( s ( n + N − . Here Φ ∗ r is the generalized Hurwitz-Lerch zeta function.29hen the dimension n > , Ω n ( k t k ) defined as Ω n ( k t k ) = F ( n − k t k ) , we have the characteristic function as follows. ψ X ( t ) = e i t T µ E [Ω n ( R t ′ Σ t )]= e i t T µ (cid:20)Z ∞ t n + N − exp( − at s )(1 + exp( − bt s )) r (cid:21) − Z ∞ Ω n ( v t T Σ t ) v n + N − exp( − av s )(1 + exp( − bv s )) r dv = e i t T µ (cid:20) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × Z ∞ F ( n v t T Σ t ) v n + N − exp( − av s )(1 + exp( − bv s )) r dv = e i t T µ (cid:20) Γ( s ( n + N − b s ( n + N − s Φ ∗ r ( − , s ( n N − , ab ) (cid:21) − × ∞ X k =0 Γ( s ( n + N − b s ( n + N − s ( t T Σ t ) k k ( n ) [ k ] k ! Φ ∗ r ( − , s ( n N + k − , ab )= e i t T µ ∞ X k =0 Γ( s ( n + N + k − s ( n + N − b ks k ( n ) [ k ] k ! Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( t T Σ t ) k . Thus, we have characteristic generators as follows: φ X ( k u n k ) = ∞ X k =0 ( − k (2 k )! γ k ( N, n, b, s ) Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( k u n k ) k ,φ X ( k u n k ) = ∞ X k =0 Γ( s ( n + N + k − s ( n + N − b ks k ( n ) [ k ] k ! Φ ∗ r ( − , s ( n + N + k − , ab )Φ ∗ r ( − , s ( n + N − , ab ) ( k u n k ) k , where γ k ( N, n, b, s ) = Γ( n ) π b s ( n + N − Γ( k + )Γ( s ( n + N + k − k + n )Γ( s ( n + N − . Appendix A.2. Proof of (7 . − (7 . Proof.
When the dimension n > , Ω n ( k t k ) defined as Ω n ( k t k ) = Γ( n ) √ π ∞ X k =0 ( − k k t k k (2 k )! Γ( k +12 )Γ( n +2 k ) ,
30e have the characteristic function as follows. ψ Y ( t ) = E ( e i t T Y ) = e i t T µ Y R ∞ t m − g m ( t ) dt × Z ∞ Ω n ( v t T Σ Y t ) N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j v N + m − j − e − av Φ ∗ r ( − e − bv , n − m j, ab ) dv = e it T µ Y " ∞ X k =0 N − X j =0 α ∗ k β j Γ( m N − j − − I , where β j = ( N − j j ! Γ( n − m + j ) b n − m + j Γ(2 r ) , α k = Γ(2 r + k )( − k k ! , α ∗ k = α k ( k + ab ) m + N − .I = Z ∞ Γ( m ) π ∞ X l =0 ( − l ( t T Σ Y t ) l v l Γ( l +12 )(2 l )!Γ( m +2 l ) × N − X k =0 ( N − k k ! Γ( n − m + k ) b n − m + k v N + m − k − e − av Φ ∗ r ( − e − bv , n − m k, ab ) dv = Γ( m ) π ∞ X l =0 ( − l Γ( l +12 )(2 l )!Γ( m +2 l ) ( t T Σ Y t ) l N − X k =0 β k Z ∞ v N + m − k − l e − av ∞ X p =0 Γ(2 r + p ) p ! ( − p e − bpv ( p + ab ) n − m + k dv = ∞ X l =0 N − X k =0 ∞ X p =0 ( − l (2 l )! Γ( m )Γ( l +12 )Γ( )Γ( m +2 l ) β k α p Γ( m + N − k + l − b N − k + l ( p + ab ) n − m + N + l ( t T Σ Y t ) l = ∞ X l =0 N − X k =0 ∞ X p =0 ( − l (2 l )! α ∗ p β k A l Γ( m N − k + l − t T Σ Y t ) l ,ψ Y ( t ) = e i t T µ Y P ∞ l =0 P N − k =0 P ∞ p =0 ( − l (2 l )! α ∗ p β k A l Γ( m + N − k + l − P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j −
1) ( t T Σ Y t ) l ,φ ( m ) ,y ( k ξ ( m ) k ) = P ∞ l =0 P N − k =0 P ∞ p =0 ( − l (2 l )! α ∗ p β k A l Γ( m + N − k + l − P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j − k ξ ( m ) k l , where A x , A x ( N, m, a, b, p, k ) = B ( m , x + ) B ( m + x, ) b N − k + x ( p + ab ) x − m +1 ,β x , β x ( N, n, m, b, r ) = ( N − x x ! Γ( n − m + x ) b n − m + x Γ(2 r ) ,α x ( r ) , α x ( r ) = Γ(2 r + x )( − x x ! , α ∗ x , α ∗ x ( N, n, a, b, r ) = α x ( x + ab ) n + N − . n > , Ω n ( k t k ) defined as Ω n ( k t k ) = F ( n − k t k ) , we have the characteristic function as follows. ψ Y ( t ) = e i t T µ Y R ∞ t m − g m ( t ) dt Z ∞ Ω n ( v t T Σ Y t ) × N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j v N + m − j − e − av Φ ∗ r ( − e − bv , n − m j, ab ) dv = e i t T µ Y " ∞ X k =0 N − X j =0 α ∗ k β j Γ( m N − j − − Z ∞ Γ( m ) π ∞ X l =0 ( t T Σ Y t ) l l ( m ) [ l ] × N − X j =0 ( N − j j ! Γ( n − m + j ) b n − m + j v N + m − j − e − av Φ ∗ r ( − e − bv , n − m j, ab ) dv = e i t T µ Y P ∞ l =0 P N − j =0 Γ( m ) π ( t T Σ Y t ) l l ( m ) [ l ] P ∞ k =0 β j α k Γ( N + m − j − b N + m − j − Φ ∗ r ( − e − bv , n − m + j, ab ) P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j − e i t T µ Y P ∞ l =0 P N − j =0 P ∞ k =0 α k β ∗ j Φ ∗ r ( − e − bv , n − m + j, ab ) P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j −
1) Γ( m ) π l ( m ) [ l ] ( t T Σ Y t ) l ,φ ( m ) ,y ( k ξ ( m ) k ) = P ∞ l =0 P N − j =0 P ∞ k =0 α k β ∗ j Φ ∗ r ( − e − bv , n − m + j, ab ) P ∞ k =0 P N − j =0 α ∗ k β j Γ( m + N − j −
1) Γ( m ) π l ( m ) [ l ] k ξ ( m ) k l , where β x , β x ( N, n, m, r ) = ( N − x x ! Γ( n − m + x )Γ(2 r ) b n − m + x , β ∗ x , β ∗ x ( N, n, m, b, r ) = β x Γ( N + m − x − b N + m − x − ,α x , α x ( r ) = ( − x Γ(2 r + x ) x ! , α ∗ x , α ∗ x ( N, m, a, b, r ) = α x ( x + ab ) n + N − . Here Φ ∗ r is the generalized Hurwitz-Lerch zeta function. Acknowledgements
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