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This research was supported in part by the National Science Foundation under Grant No. GU-2059 and the Sakko-kai Foundation. ON THE DISTRIBUTION OF A TRACE OF A MULTIVARIATE QUADRATIC FORM IN THE MULTIVARIATE NORMAL SAMPLES by Takesi Hayakawa University of North Carolina at Chapel Hill Institute of Statistical Mathematics, Tokyo Institute of Statistics Mimeo Series No. 682 Department of Statistios Univerasity of North CaroZina at Chapel Hill APRIL 1970 On the Distribution of a Trace of a Multivariate Quadratic Form in the Multivariate Normal Samples Takesi Hayakawa Department of Statistics University of North carolina Chapel Hill, North Carolina 27514 This paper considers the derivation of the p.d.f. of the trace of a noncentral multivariate quadratic form using a polynomial P (T,A) defined by Ie e the author and compares with the results of Katz et al and Ruben. The com- plex case is also discussed. KEY WORm Non-central multivariate quadratic form generating function Kronecker product representation Hermitian matrix. CLAsSIFICATIm Nlt8ER: LID This :research b1as suppoFted in part by the Nationa't Science Foundation undeF GFant No. GU-2059 and the Sa1<ko-kai Foundation. ON THE DISTRIBUTION OF A TRACE OF A MULTIVARIATE QUADRATIC FORM IN THE MULTIVARIATE NORMAL SAMPLES Takesi Hayakawa University of North Carolina at Chapel Hill Institute of Statistical Mathematics, Tokyo 1. INTRODUCTION. Recently the probability density function (p.d.f.) of latent roots of a multivariate non-central quadratic form and of a trace of it were obtained by the use of a polynomial Hayakawa [3]. P (T ,A) introduced by K However, it only shows that if we use the polynomial P (T,A), K we ean represent the p.d.f. 's in terms of a power series representation. e In this paper, we discuss another type of representation, that is, r-type rep;esentation of the p.d.f. and give a more concrete form than Hayakawa [3]. By using this representation, we can compare with the results of Ruben [8] and Katz et al [7]. We also give a p.d.f. for the case of the complex vari- abIes. 2. (tnSn) NoTATIONS PHD SM USEFlL RESULTS. real arbitrary matrices each of rank i positive definite symmetric (p.d.s.) matrix. polynomial (1) PK(T,A) T and m, and let U be A be an mxn nxn Hayakawa [3] defined a new as follows; ~(-TT')P (T,A) = 'lfmn <-.1,: f ~(-2iTU')~(-UU')C U K e Let (UAU')dU, K This research was supported in part by the National Science Foundation under Grant No. GU-2059 and the Sakko-kai Foundation. 2 where JC Ie is a partition of k = (kl'k2 , ... ,km), k into not more than m parts, i.e., = k l +. u+km, 1s a zonal polynomial of UAU' k ~ l k 2 ~ ... ~ k m ~ 0, corresponding to a partition and JC CJC(UAU') of k, James [5]. PJC(T,A) has the following properties. = (2) P (O,A) (3) PIe (T,I) n (4) IpK (T,A) I Ie (n ) 2 JC = ~ eJC (A) Ie K(In ) t HK (T), et!t(TT,)(n ) 2 Ie eK(A)/e K (1n ), where (a) and K H (T) K (a) = n • a{a+l) ••• (a+n-l) = r(a+n) rea) , is a generalized Hermite polynomial of matrix argument The generating function of P (T ,A) T. is given by Ie (5) L K P (T,A)C (W') K K k t (n) C (1 ) 2 K K m and the right hand side (R.B.S.) of (5) converges absolutely with respect to U. d(BI) onal groups H (T) K and and d{H ) 2 Oem} PIe(T,A) are the orthogonal invariant measures on the orthog- and O(n) , respectively. A detailed discussion of may be found in Hayakawa [3]. We give here some useful lemmas which will be applied to the representation of the p.d.f. of a trace of a non-central quadratic form. 3 lfr.t1A 1. (6) • • .AJ _ _J . AJ ], -'1c-2 -"'k-1 -"'k-1 where .f. D 1,2, ••• ,k and AO - PRooF: 0f for convenience. From the definition of P (T ,A), I<: we construct the generating func- tion of k co (7) l (-x) k-O k! l I<: • Wtg;') 'IT = P (T,A) I<: I e.tIt(-2iTU')et'1t.(-UU')et'1t.(xUAU') dU U det(I_xA)-(m/2) e.tIt(TT'). 4 = det(I_xA)-(m/2) ~(T(I-(I-XA)-l)T'), IlxAll where HAll means the maximum value of the absolute value of the charac- teristic roots of A. We expand the R.B.S. of (7) with respect to II xAll < 1 <1, by noting that x using x2 log de.t(I-xA) III 2 xtJt.A + T Vt.A +... + xk k k J:JtA + ••• and -1 (I-xA) • I + xA k k + x 22 A + ••• + x A + ••• Hence R.H.S. III e.xp[- ~ log det(I-xA)] e.tJt(-xTA(I-xA)-l.r,) Here we set and A O = 0J for convenience. 5 We can obtain the value of dents of k x (_l)kt P (T ,A) Ie Ie on the two sides of (7). by comparing the coeffi- By differentiating the left hand x and by setting x. 0, we have side with respect to (_l)k t P (T,A). K Ie Let us divide the R.B.S. into two parts such that R.B.S. • ~xp(g(x» • {l + g(x) + exp(f(x» 1 g2 (x) 2T • {1 + f (x) + 1 gk (x) + ••• } + ••• + kT 2~ f2 (x) + ... } , where OIl g(x) The degrees of k+l, R.B.S. x f(x)· • L j-k+l A j x.1 in the second factor are all greater than or equal to and so there is no contribution to the coefficient of Bence we need only consider the first factor. .~, g'(O) g"(O) • k x in the We have 2~, ••• , g(k)(O) • k!~, and g(O) • 0 Now we differentiate g(k) (0) • (=!a' k by definition). times the power series of k!~, I taO (~)g(l)(X)g(k-l)(X)1 xeO g(x) and set x · O. 6 In the same way we have and (gP(x»(k)\ • 0, for P x-O ~ k+l. Henee combining the results, we have which completes the proof. fxAr.9LES: k .. 1: k-2: P (T,A) ... (-l)A l l ! (2) CI - ; tJtA + .tJtTAT', P(2)(T,A).2I[~2+i~) 2 ... 21[.tJt(: A2 - TA T') + i<.tIt(~ A - TAT') }2], 7 etc. REMARK. If we set A = I. n then we have inunediately from Hayakawa [4. (18)] 2 PK(T.!) n K 3. THE PID.F. OF • l H (T) K bt K • I:-1ux'. (_I)k L(mn/2)-1 (~T') • k Let X be an mxn (m S n) matrix whose density function is given by (8) Where I: is an mxn (m S n) is an mxm p.d.s. matrix. matrix such that B is an E(X) • K nxn p.d.s. matrix and M and rank M. m. Let A be an nxn p.d.s. matrix. I.m1A 2. (PotUe:r series ztep:resentation.) p.d.f. (8). then the p.d.f. of the latent roots E-~XAX'E~ is given by Let X be distributed with A· diag(Al •••• 'A~ of 8 2 n(m 12)~(_ ~-~'I-l) (detA)~(n-m-l) ______ 2_-,-"'r.=_ (9) r (n)r (~(det2AB)(m/2) m2 r (a) _ n%m(m-l) nm PRooF: See Hayakawa [3]. m r (a _ a-l) and 2 a-l (Ai-Aj) i<j m2 where n Next we give another type (say, r-type) expression for this p.d.f. THEOREM I. rr-type representation.) Let d.f. (8), then the p.d.f. of the latent roots I""'XAX'I"'" is given by, for IIABII X be distributed with p. A .. diag(Al, ••• ,Am) of < p, 2 n(m 12)~(_ lE-~-~') ______ 2_ _......,,-_ (10) r (n)r (m)(de.t2AB)m/2 m 2 m 2 e.tJc.(-..!.. 2p The power series converges absolutely for PRooF: where •A We decompose H l y .. I""'XA~ A~ Cl. A > O. as is an orthogonal matrix of order column are positive and A)(detA)~(n-m-l) ,Ii,,;, ~ W\o'"1:f (A ' l ~ .... ,Am) m whose elements of the first and A , ••• ,Am l are latent 9 roots of yyt = I:-%XAXtt-~, and L is an mxn Stiefel matrix such that LL' = I . By inserting this decomposition into (8), we have the joint p.d. f. of A, HI m and L: (11) If we set L -+ LH , 2 H €O(n), 2 variant with respect to O(n) J!iBJ!2. Hence d(L)e.tJL(_lA).1.. I f where C= 2p LL'·I 2m Ul (LH )t = 1 and m 2 2 del) remains in- Then the integral with respect to is the same form as (5) with - pI)~ , f H • 2 then U· 12 A~L and T ... Oem) and iT-~MB-IA-~(C-l . eVL[_l 2 A~LH 2 (C- l _ I)H'L'A~ p 2 Oem) O(n) m The proof of the absolutely convergence is easily achieved by using (4), which completes the proof. NoTE. Since I Al>···>Am>O e.tJL(- Jl 2p A)(detA)~(n-m-l) n (Ai-Aj)C (21 A)dA i<j ~ 10 ... we have the following relations. (12) III e.tIt(- ~t-lm-~')(detAB/p)mn/2. I This formula can be obtained from (7) directly if we replace T with -kI-~MB-lA-~(C-l - ~)-~, and Hayakawa [3] obt'ained the p.d. f. of ~, Vtt-~XAX'I:-';; as -p, respectively. a power series We give this as Lemma 3 for comparison wi th Theorem 2. representation. l.Et+1A 3. C- l - A with x with (POtJ'er Series representation.) p.d.f. (9), then the p.d.f. of T ... ~ Let A be distributed with is given by (13) e.tIt(- tz:-~-~') T(mn/2)-1 r (mu) (de.t2AB) m/ 2 2 PRooF: r 1 <i)k k-O k I (mn) 2k L P1/i2I:-~MB-lA4-C~ ,e-1), Ie See [3]. THEOREM 2. (r-type representation.) Let f. (10), then the p.d.f. of T'" ~A A be distributed with p.d. is given by 11 (14) -1 C where C I) -P , A~BJ!z. III The series converges absolutely for PRooF: T > By applying a Fourier transform to T o. III Vc.A and inverting it, we Q.E.D. obtain (14) easily. - 4. THE p.d. f. of RELATION WITH A llUVARIATE QUADRATIC FORM. We can derive the .tItt-!xAx' by anotheT way. We denote -Xl x X III 2 • • m x III Xa III (xa l' •• "x00 ). and (Xl ,x2 ' ... ,xm) mean uct of and II , M ll2 III t and B. . , • llm • (lla l, ••• ,ll~ ), ll· (lll'll2'''.' llm)' and a covariance matrix II a M as ],11 x where and X t8B, On the otheT hand, where then Qt a x III l,2, ••• ,m. Let is distributed with denotes a Kronecker prod- .tItXAX' .. In x Ax' a·1 a a III x[I aDA] x , • m Hence the problem is reduced to the one of the univm:iate non-centTal e quadratic form. To compare with the Tesults of KDtz et a1 [7] and Ruben (8), we can assume that A is a diagonal matTix whose diagonal elements 12 The p.d.f. of tJtxAx' is derived as follows. J exp[- i<x-~)(~-leB-l)(x-~)'] 1 (15) mn (211')T • n m (dett)2'(detB)2' mn (211')T dX Tax(I QiDA)x' m n m (dett)2'(de.tAB)2' J exp[- i(E-1QiDC-l)x' T-xx' From the above integral, we derive two types of the representation. Tt£OREM 3. (POIJJer series representation.) p.d.f. (8), then the p.d.f. of T·~' Let X be distributed with is given by (16) exp[- ~(E-lQiDB-l)~,] T(mn/2)-l mn n m 22 r (1111) (dUE) 2'(detAB)2' 2 where and P (.,.) k c. PRooF: m· 1 in the definition of PK' (T ,A) ifni;. As for Lemma 2. CoROLLARY, (17) is a polynomial for The p.d.f. of T = m-lXAX' exp[- ~(t-~n-l)~,] T(mn/2)-1 mn m 22 r{mn) (de.tAB)2' 2 is given by 13 We can easily show that (17) is the same form as (13). Here we compare with the results of Kotz et a1 [7]. Katz et a1 showed the following Lemma. ~. (Kota et aZ.) with mean . 0 T x· (xl' ••• ,x ) n and covariance matrix diag(~, ••• ,an)' p.d.f. of Let a 1 ~ a 2 I ~ ••• ~ an > = (x+b)A(x+b)' and n 0, be normally distributed A be a diagonal matrix, i.e. and b · (bl, ••• ,b ), n then the is given by 00 I (18) k=O a P (_l)k (T)(n/2)+k-l 1 k 2 2r(n+ k) 2 are determined by The where the recurrence relation (20) k-l r r-O l~ with p bk (ag and b aP/k, k -r r 2 =- '2 Li=tl(1-kb t )(at ) b: -k can be obtained. of Katz et a1 should be changed to above form.) To compare with Theorem 3 and Lemma (Kotz et a1), we set B = I n in (16) and we have 1: =Im and 14 1 o yn [ - ";\1JlJ -...,... 2 I ]. (16) I QC) T (mn/2)-1 ~ f L k-O kl(T)k etIt(= Now replacing (T)L 2' -11k (1 7V'" leA-1) ~') A by 1 fiJA m and b by lJ in Lemma (Katz et 81), we have the following form. QC) ! (18) , (19) I aP (_l)k (T)(mn/2)+k-l k=O k I aPe k k:aO - 2 aP 0 = (21) , bP k = r (detA) -(m/2) e.tJt(- n ! , ri ]~ (detA) -(m/2) e:tp[ -! 1 (1-e/a ) -(m/2) 2j =1 l-e/aj i=l ij jel j = it (20)' 1 2r(mn+ k) 2 1 k (a) j=l j m ! i=l 1 2 MM') 2 (l-klJ ij )· Therefore, by comparing with (16) I and (18)', we have the following relation. (22) where k 1-1 (-1) tKPK (12M,A ) form for ai is given by Lemma 1. Hence (22) gives an explicit not involving a recurrence relation. We can also easily check that if we insert (22) into the left hand side of (19)', then we have (7). Next we compare with the f-type representation. 15 THEOREM 4. (r-type representation.) the p.d.f. (8), then the p.d.f. of Let T • ..tIr.XAX' X be distributed with is given by exp(- ~(E-l~B-l)~,) e.xp( - ..!.. T) T(m/2)-l • 2p n m. r(~)(de.tE)2(de;t2AB)2 (23) PROOF: From (15) and the proof of Theorem 3, we can easily show (23). CoRou.ARv. The p.d.f. of the T = tIt}';-~' is given by (24) We can show easily that (24) is the same form as (14). Ruben [8] gave a r-type representation of a quadratic form which is obtained in the following lemma. l.Ef.t.tA. p.d.f. of (Ruben) T = (x+b)A(x+b)' IlO kIo (25) e The { Under the same condition of Lemma (Kotz et al), the is given by a C e-(T/2p)T(n/2)+k-l (1)(n/2)+k k 2 (n/2)+k are determined by r (~+ k) 2 P 16 (26) Hence the recurrence relation a~ • k-l r b~_r a~/2k, ~. r-O k ~ 1, with can be obtained. To compare with Theorem 4 and Lemma (RubE!;ll), we set B =In in (23) and we replace A with I eA m and b t with I D 1.1 m in (26). Then we have (23) , e.xp(- 1 ":;'1.12 1.1') !!!l !! 22 f(mn) (de.tA) 2 e.xP(_"!") T(mn/2)-1 2p 2 (leA-1-lIp) • e and e.xP(_.1..) T(mn/2)-1 2p and 17 r (25) , k-O c e-(T/2p)T(mn/2)+k-1 (1)(mn/2)+k ~ 2 (mn/2)+kr (mn + k) 2 P (26) , 1 n exp[- 2 1: j=l m 1-6 l-(l-P/a )6 2 1: ~ij]· i=l j Therefore, by comparing with (23)' and (25)', we have the following relation. a: (27) = :r (detA/p)-(m/2)e;tIt(_ ~') 1 -% -1 %-1 IKP K (~ (A -lIp) ,A -lIp) and c ~, an explicit form for 1: PK(~4(A-1_I/P)%,A-l_l/p) K is given by Lemma 1. Hence (27) gives not involving a recurrence relation. We can easily check that if we insert (27) into the left hand side of (26)', then we obtain (7) by changing T and A into the appropriate variables. From (27), we have the relation E(Qka2k (L/Qlf» 2k {2k-l) II (28) \' 1 = ~ PK (72 -1 %-1 MA '2 (A -lIp) ,A -tIp), _Jr: where L X ij and = n 1 m 1: 7i1: 1: ~ij j=l j i-l x ij ' Q • m 1: n 1 1 2 1: ('8- p)xij , i eo 1 j e 1 j are independent normal variables with zero mean and unit variances, H ·(Y) 2k 1·3••• (2k-l). is an Hermite polynomial of order 2k, and (2k-1)!!. 18 NoTE. M .. 0 If we set and B· I n in (16) t then we have the p.d.f. of a central case. given in Bayakawa [2J by using the zonal polynomials. 5. THE CDPLEX M,LTIVARIATE QUADRATIC FORM. In this section, we shall state the above results for the complex Gaussian distribution studied by Goodman [1]. James [5] and Khatri [6]. Let are m, trix. and T U be mxn respectively. and We define PK (T .A) (m :S n) A be an 'll'mn K CK (UAU') where nxn positive definite Hermitian ma- as follows: m(-TT ')P (T ,A) .. (_l)k (29) complex arbitrary matrices whose rank f etIt(-i (TU '+U'f , ) )eVt(-W')e (UAl'J')dU, U· is a zonal polynomial of a Hermitian matrix K UAU'. Then we can show that (30) PK (O.A) .. [n] C (A)le (I ). KK I<: n (31) PI<: (T,I) .. HI<: (T). n :s eVt(TT')[n] I<: (32) C (A)/eK (In ), I<: where m [n] and H(T) K I<: = n (a-a+l)k' a-I a is a generalized complex Hermite polynomial of a matrix argument T. The generating function of PI<: (T ,A) is given by 19 I I (33) U(m) U(n) ~ • I I k-O Ie ebr.(-SU Atf'5'+U SU A%f'+TA-tu'S'ij')d(U )d(U ) 2 2 P (T.A)C Ie 1 2 2 1 (m ~ n) of order m and (SS') kl[n] KCK (Im) complex arbitrary matrix, n, 2 _Ie The R.B.S. of (33) converges absolutely with respect to an mxn 1 respectively, and IiK (T) l d(U ) l invariant measures over the unitary groups A detailed discussion of U U(m) and and S, U 2 S U{n), are the unitary respectively. may be found in Hayakawa [4]. lEf.tiA 2. I (34) PK(T,A) • f( where and ,.. A • O PRooF: Let given by 0, for convenience. Similar to Lemma 1. X be an mxn (mSn) is are unitary matrix d(U ) 2 and where complex matrix whose denisty function is 20 (35) where I is an whose rank 1s an nxn M is an mxm p. d. Hermite matrix, m, and B is an p.d. Hermitian matrix. nxn We denote .. .2 , xa setting = (xa l'xa 2' ••• 'xan) x = (xl'~' ••• ,xm) = xCI 8A)x I tit and m with mean x has an P and covariance mn A be l 1J 2 • • , 1J m and and 1J M = xm Let and M as X x where complex matrix p.d. Hermitian matrix. Xl X mxn 1J a 1J. = (pa l' ••• 'Pan ), (PI P2 • • Pm) , a · 1,2, ••• ,m. By we rewrite .t!lXAf1 dimensional complex Gaussian distribution Then by applying the same method as in IQ)B. the real variate case, we have the following theorem. THEOREM 5. (P01JJer series representation.) the p.d.f. (35), then the p.d.f. of Up(-p(I: -1QDB-1)}.I') (36) where and Pk ( • , •) C co A"iBA1t• is a polynomial for THEOREM 6. (r-type T =.t'JLXAI 1 Let X be distributed with is given by mn-l T m· 1 in the definition of "" P (T,A) ~pFesentation.) K Let p.s.f. (35), then the p.d.f. of T"~' X be distributed with the is given by 21 (37) where D. t- 1 ec- l m n Ip - I 0)1 and IIABII < p. Theorem 5 and 6 can be proved in the same way as the real variate case. However, since the procedure is exactly the same, we will omit it. AcKNOWl..EIlGt-£N. The author wishes to express his sincere thanks to Professor Norman L. Johnson who read carefully the original version and gave some advice. REFERENCEs. [1]. Goodman, N.R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann. Math. Statist.~ 34, 152-177. [2). Hayakawa, T. (1966). On the distribution of a quadratic form in a multivariate normal sample. Ann. Inst. Statist. Math.~ Tokyo~ 18, 191-201. (3] • Hayakawa, T. (1969). On the distribution of the latent roots of a positive definite random symmetric matrix l. Ann. Math. Statist. Math.~ Tokyo~ 21, 1-21. [4]. Hayakawa, T. (1970). On the distribution of the latent roots of a comp Z~ Wishart matri3: (Non-centraZ case). Institute of Statist. Mimeo Series No. 667. University of North Carolina at Chapel Hill. [5). James, A.T. (1964). Distribution of matrix variates and latent roots derived from normal samples. Ann. Math. Statist., 35, 475-501. [6]. Khatri, C.G. (1966). On certain distribution problems based on positive definite quadratic ft.mctions in normal vectors. Ann. Math. Stati8t.~ 37, 468-479. 22 [7]. Kotz, S., Johnson, N.L. and Boyd, D.W. (1967). Series representations of distr1butions of quadratic forms in normal variable I I. Non-central case. Ann. Math. Statist., 38, 838-848. [8]. Ruben, H. (1962). Probability content of regions under spherical normal distribution IV: the distribution of homogeneous and non-homogeneous quadratic functions of normal variables. Ann. Math. Statist., 33, 542-510.