2000 Mathematics Subject Classification: 62H10.We consider the joint distribution of the correlation coefficients for samples from multivariate standard normal distribution. Some marginal densities are obtained. Independence and conditional independence between sets of sample correlation coefficients are established

Veleva, Evelina

Bulgarian Digital Mathematics Library at IMI-BAS

Pliska Stud. Math. Bulgar. 18 (2007), 379–386STUDIA MATHEMATICABULGARICAJOINT DENSITIES OF CORRELATION COEFFICIENTSFOR SAMPLES FROM MULTIVARIATE STANDARDNORMAL DISTRIBUTIONEvelina VelevaWe consider the joint distribution of the correlation coefficients for samplesfrom multivariate standard normal distribution. Some marginal densitiesare obtained. Independence and conditional independence between sets ofsample correlation coefficients are established.1. IntroductionLet ξ = (ξ1, . . . , ξn)T be a random vector with standard normal distributionNn(~0, I), where ~0 is a zero n×1 vector, and I is the identity matrix of size n. Letξ(1), . . . , ξ(m) be a sample from ξ with size m, (m > n). Consider the matrix R,R =1 η12 . . . η1nη12 1 . . . η2n....... . ....η1n η2n . . . 1 ,where ηij is the sample correlation coefficient of the random variables ξi and ξj ,1 ≤ i < j ≤ n. The joint density of the elements ηij , 1 ≤ i < j ≤ n of the matrixR is of the form (see [4] and [7]):(1) Cn (detYn)m−n−12 ID ,2000 Mathematics Subject Classification: 62H10Key words: Multivariate normal distribution, sample correlation coefficients, independence,conditional independence.380 E. Velevawhere:• Cn is a suitable constant,(2) Cn =[Γ(m2)]n−1Γ(m−12)Γ(m−22). . . Γ(m−n+12) [Γ(12)]n(n−1)2,Γ (·) is the well known Gamma function;• Yn is a real symmetric matrix,(3) Yn =1 y12 . . . y1ny12 1 . . . y2n....... . ....y1n y2n . . . 1 ;• ID is the indicator of the set D, consisting of all points (yij , 1 ≤ i < j ≤ n)in the Euclidian real space Rn(n−1)/2 with dimension n(n− 1)/2 , for whichthe matrix Yn is positive definite.The multivariate normal (Gaussian) distribution is a basic distribution inmany models of the multivariate statistical analysis (see [1, 2, 3, 5, 6]). Thesample correlation matrix R arises naturally in the decision theory when weconsider hypothesis, connected with the correlation between the factors in theexperiment.The author in [7] proves necessary and sufficient conditions for the positivedefiniteness of the matrix Yn. These conditions allow us to obtain some of themarginal densities of the density in (1) (see [7]).2. Joint density of union of subsetsLet us denote by Vn the set Vn = {ηij , 1 ≤ i < j ≤ n}. Let S = {ηisjs , s = 1, . . . , k}be an arbitrary subset of the set of random variables Vn. To this subset Swe can attach a graph G(S) with nodes {1, 2, . . . , n} and k undirected edges{{is, js}, s = 1, . . . k}. It is easy to find that the correspondenceS → G(S)is bilateral, i.e. there exists the inverse correspondenceG → S(G).Introduce the notation fS for the joint density of the random variables be-longing to the set S. If S is the empty set ∅ we define fS = 1.Joint Densities of Correlation Coefficients for . . . 381Theorem 1. Let S1 and S2 be two subsets of the set Vn, and G1 = G(S1),G2 = G(S2) be their corresponding graphs. Denote by Ki the set of the numbersof the nodes, which are ends of edges of the graph Gi, i = 1, 2. Let K = K1∩K2.If the set K contains at least two elements and for every two elements p and qof K, p < q the random variable ηpq belongs simultaneously to S1 and S2 then(4) fS1∪S2 =fS1fS2fS1∩S2,i.e. the sets S1 and S2 are conditionally independent on the set S1 ∩ S2.P r o o f. Let us denote the number of the elements of the sets K1, K2and K by k1, k2 and k respectively. Let S∪ and S∩ be the sets S∪ = S1 ∪ S2and S∩ = S1 ∩ S2, and G∪, G∩ be their corresponding graphs, G∪ = G(S∪),G∩ = G(S∩). Re-number the vertices in the graphs G1, G2, G∪ and G∩ so thatthe nodes from the set K1 \K to have the numbers from 1 to k1 − k; the nodesfrom the set K to have the numbers from k1−k+1 to k1 and the nodes from theset K2\K to have the numbers from k1+1 to k1+k2−k. With this re-numberingwe get new graphs G?1, G?2, G?∪, G?∩and corresponding new subsets S?1 , S?2 , S?∪and S?∩of Vn, S?1 = S(G?1), S?2 = S(G?2) S?∪= S(G?∪) and S?∩= S(G?∩). The nextproposition can be found in [9].Proposition 1. Let G(S) be the corresponding graph to a subset S ⊂ Vn.Let permute the numbers of the vertices of the graph G(S) and let denote the newgraph by G?. The subset S? = S(G?) has the same joint distribution as the initialset S.According to this statement, the random variables from the set S?1 have thesame joint distribution as the random variables from the set S1, i.e.(5) fS?1 = fS1 .Analogically,(6) fS?2 = fS2 , fS?∪ = fS∪ , fS?∩ = fS∩ .It is easy to see that(7) S?∪= S?1 ∪ S?2 , S?∩= S?1 ∩ S?2 .Let us denote by U1 and U2 the sets U1 = {ηij | 1 ≤ i < j ≤ k1} andU2 = {ηij | k1−k+1 ≤ i < j ≤ k1 +k2−k} . It is obvious that S?1 ⊂ U1, S?2 ⊂ U2and U1, U2 ⊂ Vk1+k2−k. The next proposition can be found in [7].382 E. VelevaProposition 2. The joint density of the random variables from the set Vs,where s is an integer (s < n), has the form (1) with n = s.Consequently, the joint density of the random variables from the set Vk1+k2−khas the form (1). We will use the next proposition which can be found in [8].Proposition 3. Let p and q be arbitrary integers, such that 2 ≤ p ≤ n − 1and 1 ≤ q ≤ n−2. Let A and B be subsets of the set Vn, A = {ηij | 1 ≤ i < j ≤ p}and B = {ηij | q + 1 ≤ i < j ≤ n}. ThenfA∪B =fAfBfA∩B.From the last equality it follows that:(8) fU1∪U2 =fU1fU2fU1∩U2.It can be easily seen thatS?1 ∩ S?2 = {ηij | k1 − k + 1 ≤ i < j ≤ k1} .Consequently,S?1 ∩ S?2 ≡ U1 ∩ U2 .Let us integrate the two sides of the representation (8) with respect to thevariables, corresponding to the random variables from the set (U1\S?1)∪(U2\S?2).On the left we get the density fS?1∪S?2 . On the right, the variables, correspondingto the random variables from the set U1 \ S?1 appear only in the density fU1 , andthose related to the random variables from the set U2 \ S?2 appear only in thedensity fU2 . Therefore we get thatfS?1∪S?2 =fS?1 fS?2fU1∩U2=fS?1 fS?2fS?1∩S?2,whence by the equalities (5) – (7) the representation (4) follows. Theorem 2. Let S1 and S2 be two subsets of the set Vn, and G1 = G(S1),G2 = G(S2) be their corresponding graphs. Let us denote by Ki the set of thenumbers of the nodes, which are ends of edges of the graph Gi, i = 1, 2. LetK = K1 ∩K2. If the set K contains at most one element thenfS1∪S2 = fS1fS2 ,i.e. the sets S1 and S2 are independent.Joint Densities of Correlation Coefficients for . . . 383P r o o f. Let us denote the number of the elements of the sets K1, K2 andK by k1, k2 and k respectively. Let S∪ be the union S∪ = S1 ∪ S2, and G∪ beits corresponding graph, G∪ = G(S∪). We consider two cases:Case I. Let k = 0. Let us re-number the vertices in the graphs G1, G2 andG∪ so that the nodes with numbers from the set K1 can take values from 1 tok1; the nodes from the set K2 can take values from k1 + 1 to k1 + k2. Withthis re-numbering we get new graphs G?1, G?2, G?∪and corresponding new subsetsS?1 , S?2 and S?∪of Vn, S?1 = S(G?1), S?2 = S(G?2) and S?∪= S(G?∪). Accordingto Proposition 1, the random variables from the set S?1 have the same jointdistribution as the random variables from the set S1, i.e. fS?1 = fS1 . Analogically,fS?2 = fS2 and fS?∪ = fS∪. It is easy to see that S?∪= S?1 ∪ S?2 .Let us denote by U1 and U2 the sets U1 = {ηij | 1 ≤ i < j ≤ k1} andU2 = {ηij | k1 + 1 ≤ i < j ≤ k1 + k2} . According to Proposition 2, thejoint density of the random variables from the set Vk1+k2 is of the form (1). Inaccordance with Proposition 3, for U1 and U2 the representation (8) holds. Sincein this case U1 ∩ U2 = ∅ then fU1∩U2 = 1 and hence(9) fU1∪U2 = fU1fU2 .It is obvious that S?1 ⊂ U1 and S?2 ⊂ U2. Let us integrate the two sides ofthe representation (9) with respect to the variables, corresponding to the randomvariables from the set (U1 \S?1)∪ (U2 \S?2). On the left we get the density fS?1∪S?2 .On the right, the variables, corresponding to the random variables from the setU1 \S?1 appear only in the density fU1 , and those related to the random variablesfrom the set U2 \S?2 appear only in the density fU2 . Therefore we get the equalityfS?1∪S?2 = fS?1 fS?2 ,consequently it follows thatfS1∪S2 = fS1fS2 ,i.e. the two subsets of random variables S1 and S2 are independent.Case II. Let k = 1. The proof is by analogy with Case I, but here we re-number the vertices in the graphs G1, G2 and G∪ so that the nodes with numbersfrom the set K1 can take values from 1 to k1; the nodes from the set K2 cantake values from k1 to k1 + k2 − 1. After that we consider the sets U1 and U2,U1 = {ηij | 1 ≤ i < j ≤ k1}, U2 = {ηij | k1 ≤ i < j ≤ k1 + k2 − 1}. For U1 andU2, according to Propositions 2 and 3, the representation (9) holds. The rest ofthe proof is analogically to the Case I. Thus the proof is complete. 384 E. VelevaTheorem 3. Let r be an integer, r ≥ 2, and S1, . . . , Sr be a sequence ofsubsets of the set Vn, such thatSt = {ηij | i, j ∈ Mt, i < j} ,where Mt, t = 1, . . . , r are subsets of the set {1, . . . , n}. Let for all t, t = 2, . . . , rthe intersection(10) (M1 ∪ . . . ∪Mt−1) ∩Mt ,satisfies one of the next two conditions:1. The set in (10) contains at most one element;2. The set in (10) contains at least two elements. In this case for every twoelements p and q from (10), the random variable ηpq, (p < q), belongs tothe set(S1 ∪ . . . ∪ St−1) ∩ St .Then(11) fS1∪S2∪...∪Sr =fS1fS2 . . . fSrfS1∩S2f(S1∪S2)∩S3 . . . f(S1∪...∪Sr−1)∩Sr.P r o o f. The proof is by induction on r. Let r = 2, and S1, S2 be subsets ofthe set Vn of the formS1 = {ηij | i, j ∈ M1, i < j} , S2 = {ηij | i, j ∈ M2, i < j} ,where M1 and M2 are subsets of the set {1, . . . , n}. It is easy to see that theintersection S1 ∩ S2 will have the form(12) S1 ∩ S2 = {ηij | i, j ∈ M1 ∩M2, i < j} .Let us denote by G1 and G2 the corresponding graphs to the subsets S1 and S2,i.e. G1 = G(S1) and G2 = G(S2). Let Ki be the set of the numbers of the nodes,which are ends of edges of the graph Gi, i = 1, 2. It is easy to see that K1 = M1and K2 = M2. Denote by K the setK = K1 ∩K2 = M1 ∩M2 .Let the set K contain at most one element. According to Theorem 2, we havefS1∪S2 = fS1fS2 .Joint Densities of Correlation Coefficients for . . . 385From (12) it follows that the set S1 ∩ S2 is empty, whence fS1∩S2 = 1. Conse-quently, we get the representation(13) fS1∪S2 =fS1fS2fS1∩S2.Let the set K contains at least two elements and for every two elements pand q of K, the random variable ηpq, (p < q), belongs to the intersection S1∩S2.Then, according to Theorem 1 the equality (13) holds. Therefore, this Theorem 3is true for r = 2.Suppose that the Theorem 3 is true for some r, r ≥ 2. Let S1, . . . , Sr+1 be asequence of subsets of the set Vn, such thatSt = {ηij | i, j ∈ Mt, i < j} ,where Mt, t = 1, . . . , r + 1 are subsets of the set {1, . . . , n}. Let for all t, t =2, . . . , r + 1 the intersection (10) satisfies one of the conditions 1 and 2. Let usdenote by A and B the sets A = S1 ∪ . . . ∪ Sr, B = Sr+1. Let G1 and G2 be thecorresponding graphs of the subsets A and B, i.e. G1 = G(A) and G2 = G(B).Denote by Ki the set of the numbers of the nodes, which are ends of edges of thegraph Gi, i = 1, 2. It is easy to see that K1 = M1 ∪ . . . ∪ Mr and K2 = Mr+1.Let K be the intersection K = K1 ∩K2 = (M1 ∪ . . . ∪Mr) ∩Mr+1 .Suppose that the set K contains at most one element. According to Theo-rem 2, we havefA∪B = fAfB .The set A ∩B is empty, therefore fA∩B = 1. Consequently, we get the represen-tation(14) fA∪B =fAfBfA∩B.Let the set K contains at least two elements and for every two elements pand q of K, the random variable ηpq, (p < q), belongs to the setA ∩B = (S1 ∪ . . . ∪ Sr) ∩ Sr+1 .Then, according to Theorem 1 the equality (14) holds. From the induction as-sumption we have the representation (11) for the density fA. Hence, by theequality (14) the Theorem 3 follows for r + 1. Thus the proof of this Theorem 3is complete. 386 E. VelevaREFERE NCES[1] Anderson, T. W. An Introduction to Multivariate Statistical Analysis.New York, John Wiley & Sons, 1958.[2] Dempster, A. P. Elements of Continuous Multivariate Statistical Analysis.Mass., Academic Press, Reading, 1969.[3] Eaton, M. L. Multivariate Statistics. New York, John Wiley & Sons, 1983.[4] Ignatov, T. G., A. D. Nikolova About Wishart’s Distribution. An-nuaire de l’Univerite de Sofia ”St.Kliment Ohridski”, Faculte des SciencesEconomiques et de Gestion 3 (2004), 79–94.[5] Muirhead, R. J. Aspects of Multivariate Statistical Theory. New York,John Wiley & Sons, 1982.[6] Rao, C. R. Linear Statistical Inference and Its Application. New York, JohnWiley & Sons, 1973.[7] Veleva E. I. Some marginal densities of the sample correlation coefficientsof independent normally distributed random variables (in Bulgarian). An-nuaire de l’Universite de Sofia ”St.Kliment Ohridski”, Faculte des SciencesEconomiques et de Gestion 5 (2006), 353–369.[8] Veleva E. I. Conditional independence of joint sample correlation coef-ficients of independent normally distributed random variables. Proceedingsof 35-th Spring conference of the Union of Bulgarian Mathematicians 5 - 8April, 2006, Borovec, 315–320.[9] Veleva E. I., T. G. Ignatov Distributions of joint sample correlationcoefficients of independent normally distributed random variables. AdvancedStudies in Contemporary Mathematics 12(2) (2006), 247–254.Evelina VelevaDepartment of Numerical Methods and Statistics,University of Rousse ”Angel Kanchev”,Studentska str. 8,Rouse, 7017, Bulgaria,e-mail: eveleva@abv.bg

Joint Densities of Correlation Coefficients for Samples from Multivariate Standard Normal Distribution

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Joint Densities of Correlation Coefficients for Samples from Multivariate Standard Normal Distribution

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