Distributional results for L-statistics based on random vectors with

Chilean Journal of Statistics Vol. 4, No. 1, April 2013, 69–85

Distribution Theory Research Paper

Distributional results for L-statistics based on random vectors with multivariate elliptical distributions Alexander R. de Leon1 , Hamed Mahmoodian2 , and Ahad Jamalizadeh3,∗ 1

Department of Mathematics and Statistics, University of Calgary, Calgary, Canada 2 Department of Statistics, Yazd University, Yazd, Iran 3 Department of Statistics, Shahid Bahonar University, Kerman, Iran (Received: 13 September 2012 · Accepted in final form: 03 December 2012)

Abstract We consider random vectors XK×1 and YN ×1 having a multivariate elliptical joint distribution, and derive the exact joint distribution of X and L-statistics from Y, as a mixture of multivariate unified skew-elliptical distributions. This mixture representation enables us to predict X based on L-statistics from Y, and vice versa, when K = 1 and with normal and t-distributions. Our results extend and generalize previous ones in two ways: first, we consider a general multivariate set-up for which K ≥ 1 and N ≥ 2, and second, we adopt the multivariate elliptical distribution to include previous multivariate normal and t-formulations as special cases. Keywords: Linear combination · Mixture distribution · Multivariate unified skew-elliptical distribution · Order statistics · Squared-error loss. Mathematics Subject Classification: Primary 62E15 · Secondary 62H05.

1.

Introduction

A motivation for the paper is the following problem: it is usual practice to use the quiz scores to predict a student’s final test mark. More formally, if X is the final test mark and Y = (Y1 , . . . , YN )⊤ are the quiz scores, we wish to study X in terms of some linear P combination N i=1 ai Y(i) , where Y(1) < · · · < Y(N ) are the ordered quiz scores, with varying P weights ai . The results in this paper enable us to study X based on N i=1 ai Y(i) , when ⊤ ⊤ (X, Y ) follows a (N + 1)-dimensional multivariate elliptical distribution. The above problem is akin to the following that arises in electrical engineering and discussed by Wiens et al. (2006). In processing cellular phone signals from several antennae, receivers normally select only the strongest signals to reduce signal fading. Specifically, if a receiver receives N signals, only the n ≤ N strongest signals will be processed, which are then combined and used to analyze and predict the transmission system’s performance, as measured by X = (X1 , . . . , XK )⊤ , say. Various incarnations of the above problem, simplified in some way, have been studied previously by several authors. Viana (1998) and Olkin and Viana (1995) obtained the best ∗ Corresponding

author. Email: [email protected]

ISSN: 0718-7912 (print)/ISSN: 0718-7920 (online) c Chilean Statistical Society – Sociedad Chilena de Estad´ıstica

http://www.soche.cl/chjs

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A.R. de Leon, H. Mahmoodian, and A. Jamalizadeh

linear predictors in the trivariate normal distribution case with N = 2 and K = 1, where Y1 and Y2 are exchangeable such that (X, Y1 )⊤ and (X, Y2 )⊤ share a common correlation. Loperfido (2008b) considered the same set-up and derived the exact joint distribution of X and Y(2) = max(Y1 , Y2 ). Jamalizadeh and Balakrishnan (2009a) similarly derived the P exact joint distribution of X and 2i=1 ai Y(i) in the case of a trivariate normal distribution for X, Y1 , Y2 , with arbitrary covariance structure. They showed that this joint distribution is a mixture of bivariate unified skew-normal distributions and obtained a predictor for X using linear combinations of order statistics from Y1 , Y2 ; see also Balakrishnan et al. (2012) for the case of elliptical distributions. Order statistics, their linear combinations (i.e., L-statistics), and their corresponding distributions have also been similarly widely studied. Early results are provided by Gupta and Pillai (1965), Basu and Ghosh (1978), Nagaraja (1982), and Balakrishnan (1993). More recent work include Genc (2006), who derived the exact distribution of L-statistics from the bivariate normal distribution; Arellano-Valle and Genton (2007, 2008), who considered multivariate elliptical distributions; and Jamalizadeh and Balakrishnan (2008), who worked with bivariate skew-normal and skew-t distributions. Additional results are given by Jamalizadeh et al. (2009a,b), Jamalizadeh and Balakrishnan (2009b, 2010), and Loperfido (2008a). In this paper, we consider the general case of N > 2 and K > 1, and assume an elliptical joint distribution for X = (X1 , . . . , XK )⊤ and Y = (Y1 , . . . , YN )⊤ , i.e., let



X Y



  ∼ ECK+N µ , Σ , h(K+N ) ,

(1)

the (K + N )-dimensional elliptical distribution with density generator function h(K+N ) , and respective location parameter and dispersion-shape matrix

µ=



µx µy



and Σ =



Σ xx Σ ⊤ yx Σyy



,

(2)

where µ x and µ y are respective K × 1 and N × 1 location vectors of X and Y, Σ xx and Σyy are their respective K × K and N × N dispersion matrices, and Σyx is a N × K shape matrix. Note that this specification includes the multivariate normal and tdistributions, among others, and generalizes previous cases studied in the literature. With Y(N ) = (Y(1) , . . . , Y(N ) )⊤ as the vector of order statistics from Y, we derive the exact joint distribution of X and LY(N ) , where L is a P × N matrix of rank(L) = P . We show that this joint distribution is a mixture of multivariate unified skew-elliptical distributions, and obtain in the process, in the special case when K = 1 and L = a⊤ = (a1 , . . . , aN ), the best (nonlinear) predictors of X based on a⊤ Y(N ) , and of a⊤ Y(N ) based on X, under square loss function in the case of normal and t-distributions. We also present a mixture representation for the joint cumulative distribution function (CDF) of X and Y(r) , r = 1, . . . , N , in terms of bivariate unified skew-elliptical distributions. We organize the paper as follows. Section 2 presents a brief review of skew-elliptical distribution theory and presents specialized results for normal and t-distributions in the univariate and bivariate cases. The main results of the paper are then obtained in Section 3. Section 4 then concludes the paper.

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2.

Skew-Elliptical Distributions: Preliminaries

Consider the random vectors XK×1 and YN ×1 in Equation (1) above. A random vector UN ×1 is defined to have a multivariate unified skew-elliptical (SUE) distribution if and only if d

U = (Y|X > 0),

(3)

d

where “=” denotes equality in distribution, and it is understood that the inequality “X > 0” must hold for each of the components of X; we write U ∼ µy , µ x , Σ yy , Σ xx , Σ yx , h(K+N ) ). A closed-form expression for the corresponding SUEN,K (µ probability density function (PDF) is given by Arellano-Valle and Azzalini (2006) and Arellano-Valle and Genton (2005, 2010a); see also Branco and Dey’s (2001) formulation. Note that our construction of skew-elliptical distributions in Equation (3) relies on dispersion matrices which may generate an identifiability problem; see for example, Arellano-Valle and Azzalini (2006) and Arellano-Valle and Genton (2010b), for the construction of skew-normal and skew-t distributions. In such cases, the dispersion matrices should be replaced with correlation matrices. Taking h(K+N ) (u) = φ(K+N ) (u) = (2π)−(K+N )/2 exp(−u/2), u > 0, we obtain the multivariate unified skew-normal distribution (SUN) (Arellano-Valle and Genton, 2010b). Given U ∼ µ y , µ x , Σ yy , Σ xx , Σ yx ), its respective PDF fU (·) and moment-generating function SUNN,K (µ (MGF) MU (·) are given by −1 ⊤ −1 µx + Σ ⊤ φN (u; µ y , Σ yy )ΦK (µ yxΣ yy (u − µ y ); Σ xx − Σ yxΣ yy Σ yx ) , fU (u) = µx ; Σ xx ) ΦK (µ
1 ⊤ µx + Σ ⊤ exp µ ⊤ y s + 2 s Σ yy s ΦK (µ yx s; Σ xx ) MU (s) = , µx ; Σ xx ) ΦK (µ

(4)

where φQ (·; Σ ) and ΦQ (·; Σ) are the PDF and CDF of the centered Q-dimensional normal distribution with dispersion matrix Σ , respectively. In the case of the tν -kernel distribution with generator (K+N )

h

(u) =

) t(K+N (u) ν

=

ν+K+N 2

Γ Γ

ν 2




(νπ)




K+N 2

1+

u −(ν+K+N )/2 , ν

for u ≥ 0, ν > 0, where Γ(·) is the gamma function, we generate the multivariate unified skew-t (SUT) distribution, with PDF

 −1 fU (u) = TK µ x + Σ ⊤ yxΣ yy (u − µ y ); ×

tN (u; µ y , Σ yy , ν) , µx ; Σ xx , ν) TK (µ

µ y )⊤Σ −1 µy ) ν+(u−µ yy (u−µ Σxx (Σ ν+N

−1 − Σ⊤ yxΣ yy Σ yx ), ν + N



where tQ (·; Σ , ν) and TQ (·; Σ , ν) are the respective PDF and CDF of the centered Qdimensional t-distribution with scale matrix Σ and degrees of freedom ν, which we write µy , µ x , Σ yy , Σ xx , Σ yx , ν). These distributions were developed recently in as U ∼ SUTN,K (µ Jamalizadeh and Balakrishnan (2012), who obtained marginal and conditional distributions of SUE distributions. In what follows, we study univariate and bivariate SUN and SUT distributions and derive moment expressions, among others, for later use in Section 3.

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2.1

Univariate case

We now consider the univariate class of SUE distributions that arises from Equation (1) with K ≥ 1 and N = 1. In this case, Equation (2) becomes µ=



µx µy



and Σ =



Σ xx σ xy σyy



.

For U ∼ SUN1,K (µy , µ x , σyy , Σ xx , σ xy ), we get

  K X µx,i σxy,i 1 E[U ] = µy + φ √ √ µx ; Σ xx ) ΦK (µ σxx,ii σxx,ii i=1   µx,i σ xx,−ii ; Σ xx,−i|i , ×ΦK−1 µ x,−i − σxx,ii

(5)

where, for some i, σ xy =



σxy,i σ xy,−i



, µx =



µx,i µ x,−i



, Σxx =



σxx,ii σ ⊤ xx,−ii Σ xx,−i−i



,

with Σ xx,−i|i = Σ xx,−i−i − σ xx,−iiσ ⊤ xx,−ii /σxx,ii , and with φ(·) the standard normal PDF. Equation (5) is easily obtained by differentiating Equation (4) and using the following: K

X σxy,i ∂ µx + sσ σ xy ; Σ xx ) = ΦK (µ φ √ ∂s σxx,ii i=1

µx,−i − +µ



  σxy,i σ xx,−ii s σ xy,−i − ΦK−1 σxx,ii  σ xx,−ii ; Σ xx,−i|i .

µx,i + sσxy,i √ σxx,ii

µx,i σxx,ii



Next, if U ∼ SUT1,K (µy , µ x , σyy , Σ xx , σ xy , ν), we similarly obtain

!−(ν−1)/2
K 2 X µ ν ν/2 Γ ν−1 σ xy,i x,i 2
E[U ] = µy + √ ν+ √ σxx,ii µx ; Σ xx , ν) i=1 σxx,ii 2 πΓ ν2 TK (µ   √   ν−1 µx,i σ xx,−ii ; Σ xx,−i|i , ν − 1 . ×TK−1  q µ x,−i − 2 µx,i σ xx,ii ν + σxx,ii This follows from Equation (5) and from the result that for a χ2ν random variable νV with ν degrees of freedom,

h i ν ν/2 Γ ν−1
−1/2 1/2 1/2 2
E V φ(aV )ΦQ (V b; Σ ) = √ (ν + a2 )−(ν−1)/2 2 πΓ ν2  √ ν−1 b; Σ , ν − 1 , ×TQ √ ν + a2 for any real number a, any Q × 1 real vector b, and any positive definite Q × Q matrix Σ .

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Chilean Journal of Statistics

2.2

Bivariate case

Next, we consider the case N = 2 and K ≥ 1. Let Σ yy =



σyy,11 σyy,12 σyy,22



and Σ yx =



σ⊤ yx1 σ⊤ yx2



.

µy , µ x , Σ yy , Σ xx , Σ yx ), it follows from Arellano-Valle and For U = (U1 , U2 )⊤ ∼ SUN2,K (µ Genton (2010a) that U1 ∼ SUN1,K (µy,1 , µx , σyy,11 , Σ xx , σ yx1 ). In addition, we get 2·1 2·1 22·1 2·1 U2 |U1 = u1 ∼ SUN1,K (µ2·1 y (u1 ), µ x (u1 ), σyy , Σ xx , σ yx ), 2·1 where µ2·1 y (u1 ) = µy,2 + σyy,12 (u1 − µy,1 )/σyy,11 , µ x (u1 ) = µ x + σ yx1 (u1 − µy,1 )/σyy,11 , 2·1 2·1 ⊤ 22·1 = σ 2 σyy yy,22 − σyy,12 /σyy,11 , Σ xx = Σ xx − σ yx1σ yx1 /σyy,11 , and σ yx = σ yx2 − σyy,12σ yx1 /σyy,11 . It can also be shown that

  K 2·1 2·1 X µ (u1 ) σ 1  q yx,i φ  qx,i E[U2 |U1 = u1 ] = µ2·1 y (u1 ) + 2·1 2·1 2·1 µx (u1 ); Σ xx ) i=1 σ 2·1 ΦK (µ σxx,ii xx,ii ! µ2·1 x,i (u1 ) 2·1 2·1 2·1 ×ΦK−1 µ x,−1 (u1 ) − σ xx,−ii ; Σ xx,−i|i , 2·1 σxx,ii

(6)

2·1 = σ 2·1 2·1 where σxx,ii xx,ii − σyx1,i /σyy,11 , µ x,−i (u1 ) = µ x,−i + σ yx1,−i (u1 − µy,1 )/σyy,11 , σ xx,−ii = σ xx,−ii − σyx1,iσ yx1,−i /σyy,11 ,

 ⊤ σ xx,−ii − σ σ yx1,−i yx1,−i Σ2·1 − xx,−i|i = Σ xx,−i−i − σyx,11

σyx1,iσ yx1,−i σyy,11

 σ xx,−ii −

σxx,ii −

σyx1,i σ yx1,−i σyy,11

σyx1,i σyy,11

⊤

,

and where σ yx1 , σ yx2 , µ x , and Σ xx are similarly partitioned as in Section 2.1. Analogous µy , µ x , Σ yy , Σ xx , Σ yx , ν): results may be similarly obtained for U = (U1 , U2 )⊤ ∼ SUT2,K (µ U1 ∼ SUT1,K (µy,1 , µ x , σyy,11 , Σ xx , σ yx1 , ν),

(7)


22·1 2·1 σ 2·1 Σ2·1 U2 |U1 = u1 ∼ SUT1,K µ2·1 yx , ν + 1 , xx , q1 (u1 , ν)σ y (u1 ), µ x (u1 ), q1 (u1 , ν)σyy , q1 (u1 , ν)Σ
K 2·1 (ν+1)/2 q 1/2 (u , ν)Γ ν X σyx,i (ν + 1) 1 1 2·1 2
q E[U2 |U1 = u1 ] = µy (u1 ) + √ µ2·1 Σ2·1 2 πΓ ν+1 TK (µ x (u1 ); q1 (u1 , ν)Σ x , ν + 1) i=1 σ 2·1 2

xx,ii





 √  2·1 σ2·1 (u1 ) 2·1 σ xx,−ii 2·1   ν σ x,−i (u1 ) − σxixx,ii 2·1  Σ r ; q (u , ν)Σ , ν ×TK−1  1 1 xx,−i|i   2 {µ2·1 x,i (u1 )} ν + 1 + q1 (u1 ,ν)σ2·1 xx,ii

 2 −ν/2 2·1 µx,i (u1 )   × ν + 1 + ,  2·1 q1 (u1 , ν)σxx,ii 

(8)

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A.R. de Leon, H. Mahmoodian, and A. Jamalizadeh

2·1 2·1 , σ 2·1 , µ2·1 (u ), µ 2·1 (u ), σ 2·1 where σyx,i x,−1 1 xx,ii x,i 1 yx,−ii , and Σ xx,−i|i are as defined previously, and

  (u1 − µx,1 )2 . ν+ σyy,11

1 q1 (u1 , ν) = ν+1

Note that the above show that the class of SUN and SUT distributions are conveniently closed under marginalization and conditionalization.

3.

Main Results

Assume Equation (1) holds, where Σ is positive definite. In this section, we show that X and LY(N ) are jointly distributed according to a mixture of SUE distributions, where Y(N ) = (Y(1) , . . . , Y(N ) )⊤ is the vector of order statistics from Y, and L is a P × N matrix of rank(L) = P . To this end, let Y(N ) ∈ P(Y), where P(Y) = {Yi : Yi = Pi Y, i = 1, . . . , N !} is the collection of vectors Yi corresponding to the N ! different permutations of the components of Y, with Pi a N × N permutation matrix such that Pi 6= Pi′ , for all i 6= i′ . Further, let D be an (N − 1) × N difference matrix such that DY = ⊤ (Y2 − Y1 , Y3 − Y2 , . . . , YN − YN −1 ), i.e., row i of D is given by e⊤ i+1 − ei , i = 1, . . . , N , where e1 , . . . , eN are N -dimensional unit basis vectors. We give below our main result on the joint, marginal and conditional distributions of X and LY(N ) . Proposition 3.1 Assume Equation (1) holds. Then the following are true: (i) the joint CDF FX,LY(N ) (·) and joint PDF fX,LY(N ) (·) of X and LY(N ) is then given by

FX,LY(N ) (x, w) =

N! X

h πi FK+P,N −1 (x, w; Θ i ),

N! X

h πi fK+P,N −1 (x, w; Θ i ),

i=1

fX,LY(N ) (x, w) =

i=1

(K+N +P −1)

(K+N +P −1)

(9)

(K+N +P −1)

(10)

(K+N +P −1)

h h where FK+P,N −1 (·; Θ i ) and fK+P,N −1 (·; Θ i ) are the CDF and PDF of Θi , h(K+N +P −1) ), and SUEK+P,N −1 (Θ (N −1)

h η i ; Γ i ), πi = GN −1 (η (N −1)

h (N −1) ), Θ = {ξ ξ i , η i , Ω i , Γ i , Λ i }, with GN i −1 (·; Γ i ) the CDF of ECN −1 (0, Γ i , h ⊤ Σ xx Σ ⊤ yx,i L µy,i , Ω i = ξi = , η i = Dµ Σyy,i L⊤ LΣ  ⊤  Σ yx,i D⊤ ⊤ Σyy,i D , Λ i = Γ i = DΣ , Σyy,i L⊤ LΣ



µx µy,i Lµ



µ y,i = Piµ y , Σ yy,i = PiΣ yy P⊤ i , and Σ yx,i = PiΣ yx ;





,

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Chilean Journal of Statistics

(ii) the marginal CDF of LY(N ) is given by FLY(N ) (w) =

N! X i=1

(N +P −1)

h πi FP,N −1

(w; Θ 1i ),

µ y,i , Dµ µy,i , LΣ Σyy,i L⊤ , DΣ Σyy,i D⊤ , LΣ Σyy,i D⊤ }; where Θ 1i = {Lµ (iii) the conditional CDF of X given LY(N ) = w is given by FX|LY(N ) (x|w) =

N! X i=1

h(K+N −1)

q2i (w) 1·2 πi FK,N −1 (x; Θ i ),

−1) h(K+N q (w)

(K+N −1) ) (w) 2i

1·2 2i Θ1·2 where FK,N i , hq −1 (·; Θ i ) is the CDF of SUEK,N −1 (Θ

density generator function

(m) ha

with conditional

given by

ha(m) (u) =

h(m+1) (u + a) , h(1) (a)

a, u ≥ 0,

1·2 11·2 1·2 1·2 and Θ 1·2 = {ξξ 1·2 , Γ 1·2 = µx + i i (w), η i (w), Ω i i , Λ i }, with ξ i (w) ⊤ ⊤ ⊤ −1 1·2 ⊤ Σyy,i L ) (w − Lµ µy,i ), η i (w) = Dµ µy,i + DΣ Σyy,i L (LΣ Σyy,i L⊤ )−1 (w − Σ yx,i L (LΣ 1·2 ⊤ 11·2 ⊤ ⊤ −1 Σyy,i L ) LΣ Σyx,i , Γ i µy,i ), Ω i = Γi − = Σ xx − Σ yx,i L (LΣ Lµ ⊤ ⊤ 1·2 ⊤ ⊤ ⊤ −1 ⊤ Σyy,i Σyy,i L (LΣ Σyy,i L ) LΣ Σyy,i D , Λ i = Σ yx,i D − Σ yx,i L⊤ (LΣ DΣ ⊤ −1 ⊤ ⊤ ⊤ −1 Σyy,i D , and q2i (w) = (w − Lµ µy,i ) (LΣ Σyy,i L ) (w − Lµ µy,i ); L ) LΣ (iv) the conditional CDF of LY(N ) given X = x is given by

FLY(N ) |X (w|x) = h

(N +P −1)

q1 (x) where FP,N −1

N! X i=1

h(N +P −1)

q1 (x) πi FP,N −1

(w; Θ 2·1 i ),

(N +P −1)

Θ2·1 (·; Θ 2·1 i , hq1 (x) i ) is the CDF of SUEP,N −1 (Θ

) with conditional

(N +P −1) 22·1 2·1 2·1 ξ 2·1 , and Θ 2·1 , Γ 2·1 density generator function hq1 (x) i = {ξ i (x), η i (x), Ω i i , Λ i }, 2·1 µy,i + DΣ Σyx,iΣ −1 Σyx,iΣ −1 with ξ 2·1 xx (x − µ x ), xx (x − µ x ), η i (x) = Dµ i (x) = µ y,i + LΣ 22·1 −1 ⊤ 2·1 −1 ⊤ ⊤ ⊤ Σyy,i L − LΣ Σyx,iΣ xx Σ yx,i L , Γ i = Γ i − DΣ Σyx,iΣ xx Σ yx,i D⊤ , Λ 2·1 Ωi = LΣ = i ⊤ −1 ⊤ −1 ⊤ ⊤ ⊤ Σ Σ Σ Σ µ Σ µ LΣ yy,i D − LΣ yx,i xx yx,i D , and q1 (x) = (x − x ) xx (x − x ).

Proof For (i), note first that FX,LY(N ) (x, w) =

N! X i=1

P(DYi ≥ 0)P(X ≤ x, LYi ≤ w|DYi ≥ 0),

(11)

where the inequalities hold componentwise. Next, note that



      DYi ηi Γi Λ⊤ (K+N +P −1) i  X  ∼ ECK+N +P −1 , ,h , ξi Ωi LYi for i = 1, . . . , N !. For the ith term of Equation (11), we have P(DYi ≥ 0) = πi and by

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A.R. de Leon, H. Mahmoodian, and A. Jamalizadeh

Equation (3), we get

(K+N +P −1)

h P(X ≤ x, LYi ≤ w|DYi ≥ 0) = FK+P,N −1 (x, w; Θ i ),

which proves (i). Parts (ii)-(iv) follow in a straightforward manner from the mixture representation given in Equation (9) and results in Jamalizadeh and Balakrishnan (2012) on marginal and conditional distributions of SUE distributions. 

Proposition 3.1 is a generalization of Jamalizadeh and Balakrishnan (2009b) and Balakrishnan et al. (2012), and gives the relevant joint, marginal, and conditional distributions as mixtures of SUE distributions. It can also be specialized to the case of normal and t-distributions; hence, it can be viewed as extensions of Viana (1998) and Olkin and Viana h(K+N +P −1) (·; Θ ), the CDF of a SUE dis(1995). In either case, we need only replace FK+P,N i −1 tribution with parameter Θ i and density generator function h(K+N +P −1) with the CDF φ(K+N +P −1) (K+N +P −1) ) FK+P,N −1 (·; Θ i ) of a SUN distribution (with density generator function φ t(K+N +P −1)

ν in the normal case, or with the CDF FK+P,N −1 (·; Θ i ) of a SUT distribution (with den-

(K+N +P −1)

sity generator function tν ) in the case of t. It is also important to mention that the density generator function should be replaced by the characteristic generator function when Σ is singular.

3.1

Special case: results for X and a⊤ Y(N )

In this section, we consider the special case K = P = 1 with L = a⊤ = (a1 , . . . , aN ). P ⊤ Provided N i=1 ai 6= 0, the joint distribution of X and a Y(N ) is given by Equation (9) in PN Proposition 3.1. If i=1 ai = 0, then the bivariate SUE distributions in Equation (9) are singular, in which case the density generator function h(N +1) is replaced by the characteristic generator function ϕ(N +1) . Corresponding marginal and conditional distributions likewise follow from Proposition 3.1. For example, consider the exchangeable case



X Y



    2   µx σ δτ σ1⊤ (N +1) N ∼ ECN +1 µ = ,Σ = ,h , µ1N τ 2 {(1 − ρ)IN + ρ1N 1⊤ N}

p √ where τ > 0, |ρ| < 1, N |δ| ≤ 1 + ρ(N − 1), 1N = (1, . . . , 1)⊤ is the N ×1 summing vector, and IN = diag(1, . . . , 1) is the N ×N identity matrix. This equicorrelation structure for Y is found most frequently in familial studies in genetics, for example, and in animal teratology, where data arise in clusters from litters. Then, it follows from Proposition 3.1 that ) 1·2 Θ1·2 , h(N = {ξ 1·2 (w), η 1·2 (w), ω 11·2 , Γ 1·2 , λ 1·2 }, X|a⊤ Y(N ) = w ∼ SUE1,N −1 (Θ q2 (w) ), where Θ

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Chilean Journal of Statistics

with ξ

1·2

(w) = µx + τ

η

1·2



(1 − ρ)

δσ PN

PN

i=1 ai

2 i=1 ai

+ρ

P

N i=1 ai

"

2  w − µ

N X i=1

#

ai ,

# " N X (1 − ρ)Da ai , (w) = P 2 w − µ PN 2 N i=1 (1 − ρ) i=1 ai + ρ i=1 ai

ω 11·2 = σ 2 −

δ2 σ2

(1 − ρ)

PN

P

N i=1 ai

2 i=1 ai + ρ

P N

i=1 ai

2 ,

τ 2 (1 − ρ)2 Daa⊤ D⊤ P 2 , P N 2+ρ (1 − ρ) N a a i=1 i i=1 i PN δτ σ(1 − ρ)Da i=1 ai =− P 2 , PN 2 N (1 − ρ) i=1 ai + ρ i=1 ai

Γ 1·2 = τ 2 (1 − ρ)DD⊤ − λ 1·2

2

2  P w−µ N a i i=1 q2 (w) =  P 2  . PN 2 N τ (1 − ρ) i=1 ai + ρ i=1 ai

These results may be specialized as well to the normal and t-cases as generalizations of previous results by Jamalizadeh and Balakrishnan (2009b), Balakrishnan et al. (2012), Viana (1998), and Olkin and Viana (1995), among others. We now give the best (non-linear) predictors E[X|a⊤ Y(N ) = w] and E[a⊤ Y(N ) |X = x] of X and a⊤ Y(N ) , respectively, in the normal and t-cases under squared error loss. Let D =



d⊤ j D−j



=



⊤ e⊤ j+1 − ej D−j



,

(12)

where D−j is the matrix obtained from D by deleting row j, where e1 , . . . , eN are N dimensional unit basis vectors. Proposition 3.2 Suppose X and Y have a (N + 1)-dimensional normal distribution with mean µ and covariance matrix Σ . Then, under squared error loss, the best (non-linear) predictor of X based on a⊤ Y(N ) = w is N! N! h i X X 1·2 ⊤ πi ξi (w) + E X|a Y(N ) = w = i=1

i=1



N −1 X λ1·2 πi i,j q 1·2 1·2 ΦN −1 (ηη i (w); Γ i ) j=1 γ 1·2 i,jj

 ! 1·2 (w) 1·2 (w) ηi,j η i,j 1·2 1·2  ΦN −2 η 1·2 ×φ  q i,−j (w) − 1·2 γ i,−jj ; Γ i,−j|j ; (13) γ 1·2 i,jj γi,jj if, on the other hand, X and Y have a (N + 1)-dimensional tν -distribution with mean µ and scale matrix Σ , then the best (non-linear) predictor of X based on a⊤ Y(N ) = w

78

A.R. de Leon, H. Mahmoodian, and A. Jamalizadeh

becomes

h

⊤

i

E X|a Y(N ) = w =

N! X

(ν + 1)(ν+1)/2 Γ
+ √ 2 πΓ ν+1 2

ν 2




N! X

1/2

πi q2i (w, ν)   1·2 Γ , q (w, ν)Γ 2i 1·2 i i=1 i=1 TN −1 η i (w); ν+1    √  1·2 ν 1·2 (w) − ηi,j (w) γ 1·2 s η N −1 1·2 i,−jj ; i,−j γi,jj X λ1·2 {η1·2 (w)}2   i,j i,j 1·2 q TN −2  ν+1+ q2i (w,ν)γi,jj ×  1·2 γi,jj 1·2 j=1 Γ i,−j|j , ν q2i (w, ν)Γ !−ν/2 1·2 (w)}2 {ηi,j × ν +1+ . (14) 1·2 q2i (w, ν)γi,jj πi ξi1·2 (w)

All relevant quantities in Equations (13)–(14) are as defined in Proposition 3.1 and in Section 2.2; for example, η 1·2 i (w) =



1·2 (w) ηi,j 1·2 η i,−j (w)



µy,i + DΣ Σyy,i L⊤ (LΣ Σyy,i L⊤ )−1 (w − Lµ µy,i ), = Dµ

aa⊤Σ

⊤ yy,i D−j

D−j Σ yy,i a⊤Σ yy,i a  ⊤ γ 1·2 γ 1·2 i,−jj i,−jj = Γ1·2 , i,−j−j − 1·2 γi,jj

⊤ Γ 1·2 i,−j|j = D−j Σ yy,i D−j −

−

(15)

 ⊤ 1·2 γ 1·2 γ i,−jj i,−jj 1·2 γi,jj

with D given by Equation (12). The best (non-linear) predictor E[a⊤ Y(N ) |X = x] of a⊤ Y(N ) based on X = x in both the normal and t-cases is obtained by replacing “w” with “x”, changing the superscript “1 · 2” to “2 · 1”, and replacing “q2i (w, ν)” in Equation (14) with “q1 (x, ν)”. Proof Expressions (13) and (14) are straightforward from Equations (6) and (8).



Proposition 3.2 extends results in Loperfido (2008b), Jamalizadeh and Balakrishnan (2009a), and Balakrishnan et al. (2012), to the case N ≥ 2. 3.2

Special case: results for X and Y(r)

The joint CDF of X and Y(r) can be obtained from Proposition 3.1 or from Section 3.1, by taking a = er , r = 1, . . . , N . However, note that the resulting CDF involves a mixture consisting of N ! components. Evaluating the CDF may thus be a problem in practice when N is large. In this section, we present an alternative approach for deriving this joint CDF with N −1
only N r−1 < N ! terms. To do this, let 1 ≤ r ≤ N be an integer, and for integers 1 ≤ j1 < · · · < jr−1 ≤ N − 1, let Sj1 ...jr−1 = diag (s1 , . . . , sN −1 ) be a (N − 1) × (N − 1) diagonal matrix such that if jr−1 = N , then si =



1, for i = j1 , . . . , jr−2 , and i = N − 1; −1, otherwise

79

Chilean Journal of Statistics

and otherwise si =



1, for i = j1 , . . . , jr−1 ; −1, otherwise.

In particular, Sj1 ...jN −1 = IN −1 and Sj0 = −IN −1 . Further, let Yj1 ...jr−1 = (Yj1 , . . . , Yjr−1 )⊤ , and for i = 1, . . . , N , let the vector Y−i−j1 −···−jr−1 (jk 6= i, k = 1, . . . , r − 1) be obtained from Y by deleting Yi , Yj1 , . . . , Yjr−1 . Consider also the partitions Y =



Y−i Yi



, µ =



µ y,−i µy,i



, Σ yy =



Σ yy,−i−i σ yy,−ii σyy,ii



,

so that

      µx σxx σ ⊤ σyx,i X yx,−i  Y−i  ∼ ECm+N  µ y,−i  ,  Σyy,−i−i σ yy,−ii  , h(N +1)  . Yi µy,i σyy,i 

The following proposition gives the joint CDF of X and Y(r) as a mixture of SUE distribu−1
tions with only N Nr−1 < N ! terms. The resulting CDF is thus computationally simpler to evaluate and hence, more useful in practice than the corresponding CDF obtained from Proposition 3.1. Moreover, the result is technically elegant and should be of theoretical interest in its own right. Proposition 3.3 For r = 1, . . . , N , and jk 6= i, the joint CDF of X and Y(r) is given by FX,Y(r) (x, y) =

N X i=1

X

(N +1)

h πi,j1 ...jr−1 F2,N −1 (x, y; Θ i,j1 ...jr−1 ),

j1