The Distribution of the Covariance Matrix for a Subset of Elliptical Distributions with Extension to Two Kurtosis Parameters

Transcription

1 journal of ultivariate analysis 58, (1996) article no The Distribution of the Covariance Matrix for a Subset of Elliptical Distributions with Extension to Two Kurtosis Paraeters H. S. Steyn University of Pretoria, Pretoria, South Africa 1. Introduction To study the influence of deviations fro norality the class of elliptical distributions, which introduces a single kurtosis paraeter }, is often considered in statistical theory [2] to provide a wider odel than the ultivariate noral in ultivariate analysis. In this paper an expression for the oent generating function (.g.f.) of the distribution of the sapling covariance atrix for a subclass of elliptical distributions is first derived and is then generalized to include distributions containing two kurtosis paraeters. A subclass of elliptical distributions, E (0, P; }), was introduced by the present author [4] by considering the following convergent series as cuulant generating function (c.g.f.), K(t)= 1 2 (t$pt)+1 2 }[1 2 (t$pt)]2 + : r>2 A r (t$pt) r, (1a) with probability density function (p.d.f.), e (x; 0; P, })=n (x; 0, P)[ }H 2 (Q)1! +ters of higher order in the polynoial Q], (1b) where the atrix P=P _ =(\ ij ) is the population correlation atrix, t is an V 1 generating vector, A 3, A 4,..., are constants, } is a kurtosis paraeter, while n (x; 0, P) is the standardized -diensional noral with quadratic for Q=x$P &1 x. The definition of the polynoials H r (Q) are given in [4]. In particular, H 2 (Q)=Q2 &2(+2) Q+(+2). The subclass of elliptical distributions as defined by (1) contains quite a nuber Received October 26, 1993; revised Septeber AMS 1990 subject classifications: 62H15, 62H10. Key words and phrases: elliptical distributions, covariance atrix, oent generating functions, kurtosis paraeter X Copyright 1996 by Acadeic Press, Inc. All rights of reproduction in any for reserved. 96

2 ELLIPTICAL DISTRIBUTIONS 97 of well-known ebers of the elliptical class, such as the ultivariate noral, the containated ultivariate noral and the ultivariate t-distribution. An extension of E (0, P; }) to cases containing ore than one kurtosis paraeter is given in [4]. The.g.f. as defined in (1a) will be used as starting point to derive the properties of the required sapling distributions of the covariance atrix for the case of one and two kurtosis paraeters. An.g.f. approach will be used in this paper. An alternative ethod for dealing with the case of one kurtosis paraeter is given in a paper by Sutradhar and Ali [6]. 2. The Distribution of the Covariance Matrix for One Kurtosis Paraeter To derive the.g.f. of the covariance atrix for a rando saple fro the -diensional population with c.g.f. (1a), the procedure is as follows: (i) The.g.f. of the joint distribution of x i x j, ij,, 2,..., ; j=1, 2,...,, is derived by first introducing a differential operator in ters of generating variables t ij associated with x i x j respectively and which operates on the relevant.g.f. for the ultinoral case. (ii) The.g.f. of the joint distribution of the second order oents about the origin is obtained fro (i) by using straightforward probability theory and atrix differentiation. (iii) The.g.f. of the joint distribution of the eleents of the covariance atrix is obtained by applying an orthogonal transforation to the distribution derived in (ii). Starting with the case where X is ultivariate noral with p.d.f. n (x; 0, P), the.g.f. of the joint distribution of the squares and the products x i x j, ij, i, j=1, 2,...,, is given by M 1 (t)= }}} & &{ exp \ : ij=1 t ij x i x j+= C exp[&1 2Q] ` dx i, (2) where t=(t 11, t,..., t )$, C=(2?) () P & and Q=x$P &1 x= P &1 _[ P ii x i 2 +2 i<j P ij x i x j ]. Differentiating (2) with respect to t ij using D : P for the operator P &1 { : P ii +2 : P ij t ii i<j t ij=, (3)

3 98 H. S. STEYN it follows iediately that [D : P ] s M 1 (t)= }}} & &{ exp \ : ij=1 t ij x i x j+= Qs n (x; 0, P) ` Applying the result in (4), the.g.f. of the joint distribution of the squares x 2 i and the products x i x j where i and j take the values as indicated above, corresponding to the p.d.f. (1b) is given by M 2 (t)= }}} & &{ exp \ : i j=1 +higher order ters in Q] ` dx i. t ij x i x j+= n (x; 0, P)[ }H 2 (S)1! dx i =[ }H 2 (D : P) +ters of higher order in the operator D : P ]M 1 (t). (5) It is, however, well known (see [1, p. 160; 3, p. 89]) that where the atrix M 1 (t)= P &1 P &1 &T* &, T*=[(1+$ ij )], t ij =t ji, $ ij =1, i=j (4) =0, i{j. (6) Thus, using (5) and (6), the.g.f. of the joint distribution of x i x j, ij, can be obtained by applying the operator D : P on the.g.f. M 1 (t). Considering n independent vectors X=(X 1s, X 2s,..., X s )$, s=1, 2,..., n each with p.d.f. e (x; 0, P, }) as in (1b), the.g.f. of the sus of squares n s=1 x2,, 2,...,, and the products is n x s=1 isx js, ij, is given by [M 2 (t)] n, so that the.g.f. of the second-order oents about the ean, +=0, is given by M 3 (t)= 2\ t n n+&& M = {1+1 8 }H(nD 2 : P)+}}} 1\ t n n+&& =M, (7) where by (3) H 2 (nd : P)=(+2)&2(+2)(nD : P )+(nd : P ) 2 and M 1\ t n+ = P&1 } P&1 & 1 & n T*. }

4 ELLIPTICAL DISTRIBUTIONS 99 Writing now Z*=P &1 &(1n) T*, it follows on differentiation [1, p. 347] that t ii Z* & = 1 n Z* & Z* ii Z* &1 ; t ij Z* & = 1 n Z* & Z* ij Z* &1, (8) where Z* ij is the cofactor of the eleent of the i th row and j-colun in Z*. Also 2 t ij t kl Z* & = 3 n 2 Z* & Z* ij Z* kl Z* &2 & 1 n 2 Z* & ( Z* kl, ij + Z* kl, ji ) Z* &1, (9) where Z* kl, ij is the cofactor in Z* kl of the eleent in ith row and jth colun of Z*. It is clear fro Eqs. (8) and (9) that the operator t ij t kl leads to a factor Z* ij Z* kl in the first ter of the r.h.s. of (9) and to a factor ( Z* kl, ij + Z* kl, ji ) in the second ter. To write down the final for for M 3 (t) it is convenient at this stage to change the notation used in (2), where i<j. This notation gave rise to the factor 2 appearing in the second ter of (3) and will have a further accuulating effect on the second-order operators. Using the subscripts p, q, r, s, all of which can take on the values 1, 2, 3,...,, it now follows fro (8) and (9), using (3), that n_ M 3 (t)= P &1 () n Z* &() 1+1 8}(+2) & 1 4 }(+2) P &1 Z* &1 : + 1 } 8 P&2 Z* &1 : p, q=1 : p, q=1 r, s=1 P pq Z* pq P pq P rs _[3 Z* pq Z* rs Z* &1 &( Z* pq, rs + Z* pq, sr )]+}}} & n, (10) where P pq = P qp, P rs = P sr, Z* pq = Z* qp, Z* rs = Z* sr, Z*= P &1 &(1n) T*. The first factor in (10), that is, P &1 () n P &1 &(1n) T* &() n = I &(1n) T*P &() n, is the well-known.g.f. for the Wishart

5 100 H. S. STEYN W (n, (1n) P) distribution. It is clear that (10) reduces to this distribution when }=0. To proceed fro the.g.f. of the second-order oents about the ean to the.g.f. of the eleents of the covariance atrix it is observed that the p.d.f. of E (0, P; }) is a function only of the quadratic for Q=x$P &1 x. It follows, therefore, that the well-known procedure of orthogonal transforations (see, e.g., [3, p. 70]) is also applicable to this case. Thus, the.g.f. M 3 () in (10) is the.g.f. of the eleents of the covariance atrix, S=(s ij ), with the paraeters n now defined as N&1, where N is the saple size and where s ij = 1 n : N r=1 (x ir &x i)(x jr &x j). The influence on (10), if ters of third and higher degrees in (t$pt) were specified in the expressions (1a) and (1b), will be that (7) will include specific ters of higher order than the second in the differential operators. This will give rise to ters containing cofactors of the for Z* pq, rs, tu and lower order. However, the final for corresponding to (10) will contain the.g.f. of the sae Wishart distribution as first factor. It will be shown below that these ters have no influence on the standard errors of the eleents of the covariance atrix. The arginal distribution of any square subatrix S l of S located on the diagonal of S follows fro (10) but with replaced by l the rank of S l. This is seen by substituting t ij =0 for i, j>l. Taking, for exaple, l=2, the.g.f. associated with the eleents s 11, s, and s 22 follows fro (10) as M 3 (t 11, t, t 22 )=[A(t)] n{ &() 1+}&2}[A(t)]&1 1&1 n (t 11+\ t +t 22 ) & _ }[A(t)]&2 1&2 n (t 11+\ t +t 22 ) + 1 n 2 (t2 11 +\2 t2 +t t 11t 22 +\ t 11 t +\ t t 22 ) & & 1 2 }[A(t)]&1 +}}} =n, (11) where A(t)=1&(2n)(t 11 +\ t +t 22 )+(1n 2 )( )(4t 11t 22 &t 2 ).

6 ELLIPTICAL DISTRIBUTIONS 101 Hence, E( s 11 )=1 and var( s 11 )=(1n )( 2+3} ), so that for an unstandardized population with variance _ 11 (=_ 2 ) follows that var(s 1 11)= (1n)(2+3}) _ 2. Var(s 11 22) follows fro var(s 11 ) by syetry. The.g.f. for the arginal distribution of s follows fro (11) for t 11 =t 22 =0. Hence, E(s )=\ and var(s )=[1+\ 2 +}(1+2\2 ]n, so that for the unstandardized case, var(s )=_ 11 _ 22 [1+\ 2 +}(1+2\2 )]n. Siilarly, putting t 22 =0 in (11) to obtain the.g.f. of the arginal distribution of (t 11, t ) it follows fro the coefficient of t 11 t that cov(s 11, s )= (1n)(2\ +3}\ ) _ 3 1 _2 for the unstandardized case. Further, for t 2 =0 in (11), it follows for the unstandardized case that, cov(s 11, s 22 )= (1n)(2\ 2 +}(1+2\2 )] _ 11_ 22. Rewriting the well-known expressions given in [3, pp. 4142], for the variances and covariances of the asyptotic distribution of the covariance atrix under a ultivariate elliptical odel as cov(s ij, s kl )= 1 n (_ ik_ jl +_ il _ jk )+ } n (_ ij_ kl +_ ik _ jl +_ il _ jk ), () where, as usual, _ ii =_ 2 i, _ ij=\ ij _ i _ j, i{j, it is clear that the exact expressions for var(s 11 ), var(s 22 ), var(s ), and cov(s 11, s ) as derived fro (11) for the subclass defined in (1), is the sae as asyptotic expression (). Clearly, it can be seen fro the syetrical property of (10) that () also gives the exact expressions for all eleents of the covariance atrix. 3. The Distribution of the Covariance Matrix for Two Kurtosis Paraeters Dividing the rando vector X$=(X 1, X 2,..., X ) into vectors X (1) = (X 1, X 2,..., X h )$ and X (2) =(X h+1,..., X )$ with a corresponding division of the generating variable t and of the atrix P, and writing the.g.f. of the ultivariate noral in the for given in [4], Eq. (23), the c.g.f. for the case of two kurtosis paraeters follows as K(t)= 1 2 (t(1) +P &1P 11 t (2) )$ P 11 (t (1) +P &1P 11 t (2) ) + 1 } 2 1[ 1 2 (t(1) +P &1 P 11 t (2) )$ P 11 (t (1) +P &1 P 11 t (2) )] (t(2) $P 22.1 t (2) )+ 1 } 2 2[ 1 2 (t(2) $P 22.1 t (2) )] 2 +ters higher than fourth degree in t 1, t 2,..., t. (13) where P 22.1 =P 22 &P 21 P &1 11 P. The associated quadratic fors are Q 1 =X (1) $P &1 11 X(1), Q 2 =Y (2) $P & Y(2) with Y (2) =X (2) &E(X (2) X (1) =x (1) ).

7 102 H. S. STEYN It is easily seen that the for of the c.g.f. (13) is such that.g.f.'s of the distributions of subatrices or related conditional atrices of the sapling covariance atrix can be written down by using (10). For this purpose it is necessary to divide the sapling covariance atrix S into subatrices, S= _S 11 S S S 22& and to define S 22.1=S 22 &S 21 S &1 11 S. The relevant.g.f.'s, as well as soe properties, are (a) Since the arginal distribution of X (1) (obtained when t (2) =0 is e h (x (1) ; 0 } P 11, } 1 ), the.g.f. of the arginal distribution of the sapling covariance atrix S 11 associated with the vector X (1) is given by (10), where is replaced by h, } by } 1, and P by P 11. This eans that the exact variances and covariances of the eleents of S 11 (the sapling covariance atrix corresponding to P 11 ) can be written down by using (). (b) The distribution of the conditional sapling covariance S 22.1 corresponding to the atrix P 22.1 and associated with the conditional vector Y (2) is distributed independently of S 11. The.g.f. of S 22.1 follows directly fro (10) by substituting P 22.1 for P, &h for and } 2 for } and changing the relevant cofactors appearing in (10) accordingly. (c) For the case of two kurtosis paraeters an expression siilar to.g.f. (5) can be written down for the.g.f. of the distribution of the eleents of the atrix [x i x j ] of squares and products. This.g.f. will be denoted by M(t; } 1, } 2 ) and is given by M(t; } 1, } 2 )= }}} & &{ exp \ : ij=1 t ij x i x j+= n h(x (1) ; 0, P 11 ) _[ } 1H s 2 (Q 1)1!+ters of higher order] n &h (y (2) ; 0, P 22.1 )[ } 2H &h 2 (Q 2 )1! +ters of higher order] ` 1 dx i. (14) To introduce the ethod consider the three-diensional case when =3 in (14) and where the rando variables X 1, X 2, and X 3 are divided as (X 1, X 2 ) and X 3, so that P=(\ ij ),, 2, 3; j=1,2,3; P 11 =(\ rs ), r=1, 2; s=1, 2; \ pq =1 if p=q, \ P 22.1 =1& 13 &\ \ 23 \ _\ +\ 23 &\ \ 13 1&\ &\ &. (15) 2

8 ELLIPTICAL DISTRIBUTIONS 103 Further, substitute t 33 =0, so that (14) reduces to & &\ [exp(t 11x 2 1 +t x 1 x 2 +t 22 x 2 2 )] e 2(x 1, x 2 ; 0, P 11, } 1 ) _ { exp & 13_ 1\ {t x x 3+ \ 13&\ \ 23 & \ 13&\ \ 23 x 2 1&\ &\ 23&\ \ x 1&\ 2 1 x 2&= _exp {t 23_ x 2\ x 3+ \ 23&\ \ 13 & \ 13&\ \ 23 x 1 x 2 & \ 23&\ \ 13 x 1 + \ 23&\ \ 13 x 2+ x 2 + \ 13&\ \ 23 x 2 2&= x 1+ _e 1 ( y 3 ; P 22.1, } 2 ) dx 3=+ dx 1 dx 2, (16) where Y 3 =X 3 &E(X 3 X 1 =x 1, X 2 =x 2 ) =X 3 + \ 23&\ \ 13 x 2 + \ 13&\ \ 23 x 1&\ 2 1. Noting that (16) consists of a arginal factor and a conditional factor, it becoes clear that the function inside the large curly brackets is the.g.f. of the joint conditional distribution of x 1 x 3 and x 2 x 3 given x 2, x 1 1x 2, and x 2 2. The following deductions are obvious: (i) Putting t 23 =0, it is seen that the conditional distribution of x 1 x 3, given x 2, x 1 1x 2, and x 2 2, is univariate elliptical with ean and variance respectively equal to \ 23 &\ \ 13 x 1&\ 2 1 x 2 + \ 13&\ \ 23 x 2 1&\, 2 1 P 22.1 x 2 1 = { 1& _ \ 13 \ 13 &\ \ 23 \ 23 &\ \ 13 +\ 23 1&\ &= 2 x2. (17) 1 Writing X 3c for the conditional X 3, siilar expressions for the ean and the variance follow for the conditional distribution of x 2 x 3c by interchanging the subscripts ``1'' and ``2.'' It follows fro (16) that the conditional product oents of x 2 1, x 1x 2, x 2 2 with x 1x 3c and with x 2 x 3c are zero. The

9 104 H. S. STEYN conditional product oent of x 1 y 3 and x 2 y 3 follows by integrating the coefficient of t 13 t 23 in (16) and is given by P 22.1 x 1 x 2 = {1& _ \ \ 13 &\ \ 23 \ 23 &\ \ 13 +\ 23 1&\ &= x 1x 2 2. (18) It is clear fro (16), (17), and (18) that if A=(x ij ),,2,3;j=1,2,3; where the rows and coluns of A 11 correspond to the rows and coluns of P 11, then the joint distribution of x 1 x 3c and x 2 x 3c, given A 11, is elliptical, E(P 21 P &1 A 11 11, P 22.1 A 11, } 2 ). (ii) Considering now the case of n independent sets of three variables each leading to a.g.f. (14), it follows fro the.g.f. of the sus of squares and sus of products (after division by n), that the c.g.f. of the conditional distribution of n 1 n : x 1i x 3i, n 1 n : x 2i x 3i, given n 1 n : x 2 1 i, 1 n n : x 1i x 2i, 1 n n : x 2 2i, is the su of the c.g.f.'s of elliptical distributions with ean and variance given by (17) and (18). Thus the factors x 2, x 1 1x 2, and x 2 2 in (17) and (18) are replaced by the relevant ean products. The definition of the atrix A can thus be changed by replacing the eleents x i x j with (1n) n x r=1 irx jr. (iii) In the above consideration the oents about the population ean were used. However, the sae reasoning as in the case of one kurtosis paraeter holds also in the present situation. This shows that for oents about the saple eans the paraeter n should be equal to N&1, where N is the nuber of independent observations. Using now the notations S, S 11, and S 22.1, as previously introduced, it follows fro S 22.1 #s 33c =x 33 & s 22s s 11s 2 23 &2s s 13 s 23 (s 11 s 22 &s 2 ), that setting t 33 =0 in (16) did not reove the influence of s 33 on relevant eleents of the covariance atrix. (iv) Return now to expression (14) for the general case, while noting that the conditional distribution S 22.1 was already dealt with under (b) above, it follows that the values of all the generating variables t ij associated with the eleents in the rows and coluns corresponding to S 22 can be

10 ELLIPTICAL DISTRIBUTIONS 105 taken as zero. The procedure siilar to the transition fro (16) to (17) and (18)) becoes only a question of repeating the algebra in atrix notation when separating the arginal factor corresponding to S 11 fro the conditional factor corresponding to S 21. Thus using the result in (i) of this section, it follows by replacing A 11 by S 11 that the general result can be stated as: The conditional distribution of S 21 given S 11 is E(P 21 P &1 11 S 11, P 22.1 S 11, } 2 ). In this for the present result ay be copared with the known result in ultinoral theory given by Muirhead [3, p. 93]. 4. An Application As an application consider the following oents and correlation atrix, together with the conditional sapling correlations obtained fro ultitraitultiethod data with respect to 50 persons easured for two traits (T) by two different ethods (M) The four variables are X 1 :(T1M1), X 2 :(T2M1), X 3 :(T1M2), X 4 (T2M2). The distribution of the conditional sapling atrix S 22.1 given in (b) of Section 3, as well as the relations in (), are taken as basis for discussion. Cond. Moents Mean Var. Skew. Kur: # 2 Correlation: X 1 X 2 X 3 X 4 correlation: X &0.04 & r.34 =0.315 X & r 34. =0.373 X &0.31 & X & The relevant questions concern the degree to which the interdependence between traits (ethods) can be attributed to ethod (trait). Although the underlying odel to be discussed contains two kurtosis paraeters, the conditional sapling atrix S 22.1 depends on only one kurtosis paraeter, so that using results in Muirhead [3, Chap. 5], the correlation structure can be treated in stead of the covariance structure. Estiates of the two kurtosis paraeters in (13) follow fro equating the calculated arginal kurtosis paraeters # 2 3 to the expected kurtosis paraeters of the arginal distributions of X 1, X 2, X 3, X 4 by using the relations siilar to those given in [4, p. 9, Eq. (20]. The four expected kurtosis paraeters (one-third of the relevant # 2 ) follows directly fro the relevant four coefficients of t 4 18, t 4 28, t 4 38, and t 4 48 in (13) in ters of the two paraeters } 1 and } 2 and the population correlations. Substituting

11 106 H. S. STEYN the calculated arginal kurtosis as well as the correlations as given in the above table, four linear oent equations in two population paraeters } 1 and } 2 can be written down. A solution to the estiation proble in a siilar case by cobining the ethod of oents and of weighted least squares, was discussed in detail by the present author in [5]. However, in the present case, where the expected values of the coefficients of t 4 18 and t 48 are both equal to } 2 1 an estiate }^ 1=&( )6= &0.092 sees feasible. Using this estiate of } 1 in the two equations obtained fro the coefficients of t 48 and 3 t4 48, two slightly different estiates &0.628 and &0.606 for } 2 follow, giving an average estiate }^ 2= & Aongst others, the conditional correlation between X 3 and X 4, that is, r=r 34. can now be transfored [3, p. 159] to z= 1 2 log [(1+r)(1&r)] and considered as noral variate with variance 1(n&2)+} 2 (n+2), where n+1 is the saple size. Applying this result it follows that the value r 34. =0.373 in the above table is significant. A siilar result is obtained on considering the correlation between X 1 and X 2, conditional on X 3 and X 4. References [1] Anderson, T. W. (1958). An Introduction to Multivariate Statistical Analysis. Wiley, New York. [2] Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale paraeter generalization. Sankhya A [3] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York. [4] Steyn, H. S. (1993). On the proble of ore than one kurtosis paraeter in ultivariate analysis. J. Multivariate Anal. 44 No [5] Steyn, H. S. (1976). On the ultivariate Poisson noral distribution. J. Aer. Statis. Assoc. 71, No [6] Sutradhar, B. C., and Ali, M. M. (1989). A generalization of the Wishart distribution for the elliptical odel and its oents for the ultivariate t odel. J. Multivariate Anal