Administration, Department of Statistics and Econometrics, Sofia, 1113, bul. Tzarigradsko shose 125, bl.3, Bulgaria,

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1 Adanced Studie in Contemorary Mathematic, (006), No, DISTRIBUTIONS OF JOINT SAMPLE CORRELATION COEFFICIENTS OF INDEPEENDENT NORMALLY DISTRIBUTED RANDOM VARIABLES Eelina I Velea, Tzetan G Ignato Roue Unierity, Deartment of Numerical Method and Statitic, Roue, 707, Studenta 8, Bulgaria, eelea@abbg Sofia Unierity St Kliment Ohridi, Faculty of Economic and Buine Adminitration, Deartment of Statitic and Econometric, Sofia, 3, bul Tzarigrado hoe 5, bl3, Bulgaria, ignato@febuni-ofiabg Abtract The author inetigate when et of amle correlation coefficient of the multiariate normal (Gauian) ditribution are deendent or indeendent 000 Mathematic Subject Claification Ditribution of tatitic 6H0 Keyword and hrae Marginal denitie, Multiariate normal (Gauian) ditribution, Samle correlation coefficient Introduction T Let ξ = ( ξ, ξ ) be a random ector (here and ubequently the oeration tranoe of a matri i deignated with uer inde T) with ditribution N ( 0, I), where 0 i a zero р х ector, and I i the identity matri of ize р Let ξ,, ξ be a amle from ξ with ize n Conider the matri R, where i j () ( n) R =, i the amle correlation coefficient of the random ariable ξ i and i < j The joint denity of the element, i < j of the matri R i (ee [4] and [7] ): i j ξ j,

2 () C det ( n ) I A, where: - I A i the indicator of the et A, coniting of all oint { i j, i < j } in for which the ymmetric matri in () i oitie definite; - i a contant, C R ( ) / () C = n Γ n n n + Γ Γ Γ Γ ( ), where Γ() i the well nown Gamma function The multiariate normal (Gauian) ditribution i a baic ditribution in many model of the multiariate tatitical analyi ( ee [,,3,5,6] ) The amle correlation matri R arie naturally in deciion theory when we conider hyothei, concerning the correlation between the factor in the eeriment Neceary and ufficient condition for the oitie definitene of the matri in () are obtained by Velea ([7]) Thee condition allow to be obtained ome of the marginal denitie of the denity () (ee [7]) The goal of thi aer i to attach to eery ubet S of the et (3) V = { ν, i < j } a ecial undirected grah, the form of which will gie u an information about tochatic deendence or indeendence of the random ariable belonging to S Rereentation with grah Let S = {, =,, } i j be an arbitrary ubet of the et of random ariable V To thi ubet S we can attach a grah G (S) with node {,,, } and undirected edge { { i, j }, =,, } It i eay to find that the correondence S G(S) i one-to-one, ie it eit the inere correondence G S(G)

3 Theorem Let G G(S) be a grah, correonding to a ubet S V from (3) If we change the number of two ertice of the grah G(S), then the correonding ubet S V of the new grah G G(S ) ha the ame ditribution a the initial ubet S Proof: We begin with the roof of Theorem for the articular cae S = V Let the number of the node and of the grah G(S) be interchanged From formula () it i eay to ee that the et V = S(G ) ha the ame joint ditribution a the et V = S(G), where G i a grah obtained by grah G with interchanging the number of node and Really, thi change i equialent to interchanging in formula () the th and th column and alo the th and th row of the matri From the well nown roertie of determinant we hae: det = det Conequently, the joint denity of V = S(G) and the joint denity of V = S(G ) coincide Taing into account the lat tatement, we can conclude that Theorem i true in the general cae (when S V ), becaue from the coincidence of the joint denitie of V and V follow the coincidence of the correonding marginal denitie of S and S Corollary Let G(S) be the correonding grah to a ubet S V Let ermute the number of the ertice of the grah G(S) and let denote the new grah by G The ubet S = S(G ) ha the ame joint ditribution a the initial et S Proof: Eery ermutation of {,,, } can be rereented by roduct of tranoition of air Uing the Theorem and the aboe tatement the roof of the Corollary i comleted 3

4 Theorem Let G(S) be the correonding grah to a ubet S V, coniting of edge and not containing a circuit Then the ariable of the et S are mutually indeendent Proof: We will roe thi Theorem uing the mathematical induction Let = 3 We hae to roe that the random ariable, are airwie indeendent 3, D к Introduce the ymmetric matrice, (4) D к =, =,,, D = 3 ariable Uing () with = 3 the joint denity function 3,, 3 ha the form: f, 3, 3 (, 3, 3) of random ( n 4) 3 f,, (, 3, 3) 3 = C 3 3det 3 = ( ( )) ( n C 4) 3 det D For breity, we will denote further det ( ) = Becaue the determinant of the matri element, D 3 i a ymmetric function with reect to it 3 = D, 3 3 it i enough to roe the indeendence of only one air, for intance of and 3 We will ue the formula for the joint denity of the random ariable, i < j and of the ariable, which ha been obtained in [7] Thi denity i: ν ν (5) f,, (,, ) i < j i < j where n + Γ Γ n = C I A n Γ D, I i the indicator of the et A, coniting of all oint in R, for which the A matrice in (5) D and are oitie definite ( )( ) / + 4

5 If we integrate the joint denity f,, ) oer R with reect to the, ( 3 3, 3 3, 3 ( 3 ariable, we will get the denity f, ), which i of the form (5) for =, ie 3 3 n Γ Γ (6) f C (, ) = ( ) ( ) I, A n Γ = C ) ( 3 ) I A (, where I A i the indicator of the ubet of, for which, and 3, It ha been roed in [7] that all ariable in the et V hae the ame ditribution with denity R ( ) (7) f ( ) = C ( ), (, ) So from (6) and (7) it follow that f (, 3 ) = f ( ) f ( 3 ),, 3 3 ( ) which imlie that the two random ariable and 3 are indeendent Aume the tatement i true for, where i an integer, 3 and let = + Conider the grah G(S), correonding to a ubet S V coniting of edge and not containing a cloed ath If we uoe that there i a node in the grah G(S) that i not connected ia an edge, then thee edge will join no more than node and conequently they will form a circuit Therefore, each of the node i connected with at leat one of the ret node There i a node connected with eactly one of the ret node Suoing the ooite that at leat two edge go out from each node, would lead to a contradiction with the total number of the edge Let i be the number of a node, from which eactly one edge goe out Interchanging the number of the node i and + of G we get a new grah G with correonding new ubet S, S V According to Theorem, the ditribution of the random ariable from S and S coincide Thu in further conideration we can ue the tatement that from the node + eactly one edge goe out Let thi edge be { j, + }, j Interchanging the number of the node j and by the imilarly way a aboe we can tate that without lo of generality S,, } { }, { + + = + ie from the ariable rereented by the element of the lat column of the matri ν belong to S Conequently + S { +,, i < j } D + only 5

6 Uing (5) for р = + we hae f, i< j, + (, i < j, + ) = C + n Γ Γ n Γ D n + + I A = C D n C n ( ) 3 I, + A where i the indicator of the ubet of R, for which the matri D i oitie definite and I A ( ) / + (,) + The lat formula imlie that the random ariable + i indeendent of the et { ν, i < j } Hence i indeendent of eery ubet of { ν, i < j } Therefore + to roe the tatement it i enough to roe that the ret ariable in S indeendent Since S \{ + } {, i < j } are mutually the mutual indeendence of the ret ariable in The roof of Theorem i comleted S follow from the induction aumtion Theorem 3 Eery random ariable from the et { ν, i < j } are deendent Proof: Suoe the ooite of the Theorem that there i a ubet S, S V, coniting of р random ariable: ϑ,,ϑ, which are mutually indeendent All the ariable, i < j are identically ditributed with denity (7), conequently the joint denity of ν the ariable ϑ, i =, will be i, ϑ,ϑ, i= (8) f ( y,, y ) = f ( y ) f ( y ) = ( C ) ( y ), y i (, ), i =,, i Introduce column-ector, by T =,, ), =,, ( 6

7 It ha been roed in [7] that the ymmetric matri diagonal i oitie definite if and only if D in (4) with unit on the main < (9) D, T for =,, The ector, =,, contain all the ariable, correonding to the ariable j, and conequently the ariable a well correonding to ϑ,ϑ The ν, i <, number of the random ariable ϑ,,ϑ i and the number of the ector,, i - Therefore at leat one from the ector, =,, will contain at leat two of the ariable correonding to the indeendent ariable ϑ,,ϑ Let thi be The condition (9) for = i < (0) D The inequality (0), dicued with reect to the ariable T, i =,, -, articiating in, reent a - dimenional ellioid It i well nown that the rojection of any ellioid i an ellioid in maller dimenional ace Conequently it i not oible the rojection of the ellioid from (0) to be (,) (,) Thi contradiction comlete the roof Theorem 4 Let G G(S) be a grah, correonding to a ubet S V from (3) If G contain at leat one cloed ath (circuit), the random ariable from S are deendent In the ooite cae they are mutually indeendent Proof: Let G contain or more edge then urely they form at leat one cloed ath From Theorem 3 it follow that the ariable from S are deendent Now let G hae le than edge and contain a cloed ath (circuit) Then there eit at leat one node of the grah G, which i not connected ia an edge Let thee node be in number, < < Without lo of generality we can aume that thee node are numbered from + to Each of the node with number from to i connected ia an edge and the edge form at leat one cloed ath Conequently the number of the edge in the grah i at leat It ha been roed in [7] that the joint denity of the ariable, i < j, where i an integer, < i equal to the denity in () for = Therefore the joint denity of the ariable, i < j will be of the form () for = Conequently, according to ν Theorem 3 the ariable from the et S are deendent The econd art of the tatement follow from Theorem and from the fact that the maimal number of edge of a grah with node, not containing a circuit i ν i 7

8 Reference [] Anderon, T W An Introduction to Multiariate Statitical Analyi New Yor, John Wiley & Son, 958 [] Demter, A P Element of Continuou Multiariate Statitical Analyi Ma, Academic Pre, Reading, 969 [3] Eaton, M L Multiariate Statitic New Yor, John Wiley & Son, 983 [4] Ignato, T G, A D Nioloa, About Wihart Ditribution, Annuaire Uni Sofia, Fac of Econ & Bu Admin, 3(004), [5] Muirhead, R J Aect of Multiariate Statitical Theory New Yor, John Wiley & Son, 98 [6] Rao, C R Linear Statitical Inference and It Alication New Yor, John Wiley & Son, 973 [7] Velea, E I Some marginal denitie of the amle correlation coefficient of indeendent normally ditributed random ariable (in Bulgarian) Annuaire Uni Sofia, Fac of Econ & Bu Admin, 5(006),

A New Criterion for Meromorphic Multivalent Starlike Functions of Order γ defined by Dziok and Srivastava Operator

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