REVSTAT – Statistical Journal Volume 11, Number 2, June 2013, 121–134 THE EXACT JOINT DISTRIBUTION OF CONCO- MITANTS OF ORDER STATISTICS AND THEIR ORDER STATISTICS UNDER NORMALITY Authors: Ayyub Sheikhi – Department of Statistics, Shahid Bahonar University of Kerman, Iran [email protected] Mahbanoo Tata – Department of Statistics, Shahid Bahonar University of Kerman, Iran [email protected] Received: December 2011 Revised: May 2012 Accepted: August 2012 Abstract: • In this work we derive the exact joint distribution of a linear combination of con- comitants of order statistics and linear combinations of their order statistics in a multivariate normal distribution. We also investigate a special case of related joint distributions discussed by He and Nagaraja (2009). Key-Words: • unified skew-normal; L-statistic; multivariate normal distribution; order statistic; concomitant; orthant probability. AMS Subject Classification: • 62G30, 62H10. 122 A. Sheikhi and M. Tata Joint Distribution of Concomitants and their Order Statistics 123 1. INTRODUCTION Suppose that the joint distribution of two n-dimensional random vectors X and Y follows a 2n dimensional multivariate normal vector with positive definite covariance matrix, i.e. X µ (1.1) ∼ N µ = x , = xx xy Y 2n µ T y Pxy Pyy ! X P P where µx, µy are respectively the mean vectors and xx, yy are the positive definite variance matrices of X and Y, while xy is their covariance matrix. Let T P P X(n) = (X1:n, X2:n,..., Xn:n) be the vector of order statistics obtained from X T P and Y(n) = (Y1:n, Y2:n,..., Yn:n) be the vector of order statistics obtained from T Y. Further, let Y[n] = (Y[1:n], Y[2:n],..., Y[n:n]) be the vector of Y -variates paired with the order statistics of X. The elements of Y[n] are called the concomitants of the order statistics of X. Nagaraja (1982) has obtained the distribution of a linear combination of order statistics from a bivariate normal random vector where the variables are exchangeable. Loperfido (2008a) has extended the results of Nagaraja (1982) to elliptical distributions. Arellano-Valle and Genton (2007) have expressed the ex- act distribution of linear combinations of order statistics from dependent random variables. Sheikhi and Jamalizadeh (2011) have showed that for arbitrary vectors T T T a and b, the distribution of (X, a Y(2), b Y(2)) is a singular skew-normal and carried out a regression analysis. Yang (1981) has studied the linear combina- tion of concomitants of order statistics. Tsukibayashi (1998) has obtained the joint distribution of (Yi:n,Y[i:n]), while He and Nagaraja (2009) have obtained the joint distribution of (Yi:n,Y[j:n]) for all i, j = 1, 2,..., n. Goel and Hall (1994) have discussed the difference between concomitants and order statistics using the n sum i=1 h(Yi:n − Y[i:n]) for some smooth function h. Recently much attention has been focused on the connection between order statistics and skew-normal P distributions (see e.g. Loperfido 2008a and 2008b and Sheikhi and Jamalizadeh T T 2011). In this article we shall obtain the joint distribution of a Y(n) and b Y[n], T T n where a = (a1, a2,..., an) and b = (b1, b2,..., bn) are arbitrary vectors in R . Since we do not assume independence, our results are more general than those of He and Nagaraja (2009). On the other hand, He and Nagaraja (2009) have not assumed normality. The concept of the skew-normal distribution was proposed independently by Roberts (1966), Ainger et al. (1977), Andel et al. (1984) and Azzalini (1985). The univariate random variable Y has a skew-normal distribution if its distribu- tion can be written as y − µ (1.2) f (y) = 2ϕ y; µ, σ2 Φ λ y ∈ R Y σ 124 A. Sheikhi and M. Tata where ϕ .; µ, σ2 is the normal density with mean µ and variance σ2 and Φ(.) denotes the standard normal distribution function. Following Arellano-Valle and Azzalini (2006), a d-dimensional random vec- tor Y is said to have a unified multivariate skew-normal distribution (Y ∼ SUNd,m (ξ, δ, Ω, Γ, Λ)), if it has a density function of the form (1.3) T −1 T −1 Φm δ + Λ Ω (y − ξ); Γ − Λ Ω Λ f (y)= ϕ (y; ξ, Ω) y ∈ Rd Y d δ Γ Φm ( ; ) where ϕd(., ξ, Ω) is the density function of a multivariate normal and Φm(.; Σ) is the multivariate normal cumulative function with the covariance matrix Σ. Γ ΛT If Σ∗ = is a singular matrix we say that the distribution of X is Λ Ω singular unified skew-normal and write SSUNd,m(ξ, δ, Ω, Γ, Λ). For more de- tails see Arellano-Valle and Azzalini (2006) and Sheikhi and Jamalizadeh (2011). In Section 2, we show that for two vectors a and b, the joint distribution of T T a Y(n) and b Y[n] belongs to the unified multivariate skew-normal family. We also discuss special cases of these distributions under the setting of independent normal random variables. Finally, in section 3 we present a numerical application of our results. 2. MAIN RESULTS Define S(X) as the class of all permutation of components of the ran- (i) dom vector X, i.e. S(X)= {X = PiX; i = 1, 2,..., n!}, where Pi is an n × n permutation matrix. Also, suppose ∆ is the difference matrix of dimension T T (n − 1) × n such that the ith row of ∆ is en, i+1− en, i, i = 1,...,n − 1, where e1, e2,..., en are n-dimensional unit basis vectors. Then ∆X = (X2 − X1, X3 − T X2,..., Xn − Xn−1) . (See e.g. Crocetta and Loperfido 2005). Further, let X(i) and Y(i) be the ith permutation of the random vectors X (i) (j) and Y respectively. We write Gij(t, ξ, )= P ∆X ≥ 0, ∆Y ≥ 0 . P ∆ Theorem 2.1. Suppose the matrix aT is of full rank. Then under bT T T T the assumption of model (1.1) the cdf of the random vector a Y(n), b Y[n] is the mixture n! n! F T T ( y , y )= F ( y , y ; ξ , δ , Γ , Ω , Λ ) G (t, ξ, ) a Y(n), b Y[n] 1 2 SUN 1 2 ij ij ij ij ij ij i j X=1 X=1 X Joint Distribution of Concomitants and their Order Statistics 125 where FSUN (. ; ξij, δij, Γij, Ωij, Λij) is the cdf of unified multivariate skew- normal with T (i) (i) (ii) T (ij) T a µy ∆µx ∆ xx ∆ ∆ xy ∆ ξ = , δij = , Γij = , ij T (i) (j) ∆ (jj) ∆T b µy ! ∆µy ! P Pyy ! P T T (ii) T (ij) (ii) (ij) a yy a a yy b ∆ xy a ∆ xy b Ωij = and Λij = bT (jj) b ∆ (ij) a ∆ (jj) b P Pyy ! Pyy Pyy ! (i) (j) where µx and µy areP respectively the mean vectors ofP the ith permutationP of the random vector X and the jth permutation of the random vector Y and (ii) (i) (jj) (j) (ij) (i) (j) xx = V ar(X ), yy = V ar(Y ) and xy = Cov(X , Y ). P P P Proof: We have T T F T T (y , y ) = P a Y ≤ y , b Y ≤ y a Y(n), b Y[n] 1 2 (n) 1 [n] 2 n! n! T (i) T (j) (ij) (ij) = P a Y ≤ y1, b Y ≤ y2|A P A i j X=1 X=1 where A(ij) = {∆X(i) ≥ 0, ∆Y(j) ≥ 0}. Since ∆X(i) ∆ 0 0 X(i) ∆Y(j) 0 ∆ 0 (2.1) = Y(j) , aT Y(i) 0 0aT Y(i) bT Y(j) 0 bT 0 3n×1 2n×1 2n×3n the full rank assumption implies nonsingularity of the matrix on the right hand side of (2.1). Furthermore, ∆X(i) ∆Y(j) aT Y(i) bT Y(j) (i) (ii) T (ij) T (ii) (ij) ∆µx ∆ xx ∆ ∆ xy ∆ ∆ xy a ∆ xy b ∆µ(j) ∆ (jj) ∆T ∆ (ij) a ∆ (jj) b ∼ N y , P Pyy Pyy Pyy . 2n T (i) T (ii) T (ij) a µy a a a b P P yy P yy bT µ(j) bT (jj) b y P Pyy P Now, similar to Sheikhi and Jamalizadeh (2011), we immediately conclude that T T (i) T (j) (i) (j) a Y , b Y | ∆X ≥ 0, ∆Y ≥ 0 ∼ SUN2, 2(n−1)(ξij, δij, Γij, Ωij, Λij) . This proves the Theorem. 126 A. Sheikhi and M. Tata T Remark 2.1. If the rank of the matrix ∆, aT , bT is at most n − 1, the T T T joint distribution of a Y(n), b Y[n] is a mixture of a unified skew-normals and a singular unified skew-normals. In this section we assume that the matrix T ∆, aT , bT is of full rank. A special case will be investigated later in the paper. Let (Xi,Yi), i = 1, 2,..., n be a random sample of size n from a bivariate 2 2 normal N2 µx, µy, σx, σy, ρ , then the model (1.1) reduces to the following: X µx1n xx xy (2.2) ∼ N2n µ = , = Y µy1n yy X P P 2 2 T where xx = σxIn, yy = σyIn and xy = ρσxσy1n1n whereP ρ is the correla- tion coefficient between X and Y. P P P The following corollary describes the joint distribution of a linear combi- nation of concomitants of order statistics and a linear combination of their order statistics under the independence assumption.