The Generalized Integer Gamma DistributionA Basis for Distributions in Multivariate Statistics

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1 Journal of Multivariate Analysis 64, 8610 (1998) Article No. MV The Generalized Inteer Gaa DistributionA Basis for Distributions in Multivariate Statistics Carlos A. Coelho Universidade Te cnica de Lisboa, Lisbon, Portual Received Auust 6, 1996 In this aer the distribution of the su of indeendent 1 variables with different scale araeters is obtained by direct interation, without involvin a series exansion. One of its articular cases is the distribution of the roduct of soe articular indeendent Beta variables. Both distributions are obtained in a concise anaeable for readily alicable to research work in the area of ultivariate statistics distributions. The exact distribution of the eneralized Wilks' 4 statistic is then obtained as a direct alication of the results Acadeic Press AMS 1991 subect classification nubers 6E15, 6H10. Key words and hrases indeendent Gaa variables; different scale araeters; inteer shae araeters; indeendent Beta variables; Wilks' Labda; likelihood ratio statistic. 1. INTRODUCTION The distribution of the su of indeendent Gaa rando variables with different scale araeters and the distribution of the roduct of articulandeendent Beta rando variables has been studied by a few authors. All the results have been obtained under the for of series exansions. Indeed, Kabe [7], by invertin the characteristic function of a linear cobination with all ositive coefficients of indeendent Gaa variates, obtained its distribution in ters of a hyereoetric series. Tretter and Walster [10] and Nandi [9] exressed the distribution of the roduct of articulandeendent Beta variates as a ixture of incolete Beta distributions, involvin a series reresentation. Guta and Richards [6] by exandin the exonential ter and usin ter by ter interation resented the distribution of the su of indeendent Gaa variables with different araeters also as a series exansion. Let X Coyriht 1998 by Acadeic Press All rihts of reroduction in any for reserved. Y i t1(, * i ),, * i >0 86

2 GENERALIZED GAMMA DISTRIBUTION 87 be a short notation for the fact that the robability density function (.d.f.) of Y i is a Gaa distribution with shae araeter and scale araeter * i, this is f Yi ( y i )= * i 1( ) y &1 e &* i y i i, yi >0,, * i >0. (1) Now let Y 1,..., Y n be indeendent rando variables havin distributions iven by (1), with * 1 =}}}=* n =*, and let Y=Y 1 +}}}+Y n. Then it is a well known fact that Yt1(r, *) with r=r 1 +}}}+r n. In this aer we will obtain the distribution of Y=Y 1 +}}}+Y n under the situation where all the * i 's are different, without involvin a series exansion, as lon as all the 's are inteer. We will call such distribution a Generalized Inteer (GI) Gaa distribution and after soe silifications it is resented in a concise and easily anaeable for. Not only is this distribution a eneralization of the Gaa distribution, but also one of its articular cases is the distribution of the loarith of the roduct of an even nuber of articulandeendent Beta rando variables. Therefore such articular GI Gaa distribution ay be used to obtain the distributions of a nuber of ultivariate statistics.. A PRELIMINARY RESULT A key result for the develoents ahead is the followin. Result 1. Fonteer and q, and for z>0, z 0 (z&x) q&1 x &1 e &kx dx =(&1) q (q&1)! \ &1 q&1 +( &1)! =0 =0 (&1) (&1)! a &, q z k e&kz! &q&+1+ (q&1)! q&1& a q&, z k &q&+1!

3 88 CARLOS A. COELHO where with =(&1)! (q&1)! {(&1)q \ a &+1, q ( &1)! z &1 k + &q& e&kz q +(&1) \ a &&q+= q&+1, z &1 (&k) (for k{0) (&1)! \ = z+q&1 a, q (+q&1) for k=0 + a, = i =1 i i &1 =1 = \+& &1 i 3 }}} i =1 i i 1 =1 + = \ +& 1= a i, &1 =a,,, 1, () &1 + for all i1. a i,1 =a 1, i =1, The above result is obtained on interatin successively by arts. 3. TWO USEFUL DISTRIBUTIONS Usin Result 1 we ay easily obtain the distribution of the su of indeendent Gaa rando variables with different scale araeters and inteer shae araeters. We have then the followin theore. Theore 1. Let Y i t1(, * i ),, be two indeendent Gaa rando variables. Then, if (, ) are inteer, the.d.f. of Z=Y 1 +Y is, for z>0, f Z (z)= \ * i + (&1) i\ S&r (for * 1 {* ) a ri &+1, S& ( &1)! z &1 (* i &L) &S+ e&* i z, \ = *S 1(S) zs&1 e &*z for * 1 =* =* +, (3)

4 GENERALIZED GAMMA DISTRIBUTION 89 where S=r 1 +r, L=* 1 +* and the a ri & +1, S& are defined as in (). Proof. Fro (1), the distribution of Z=Y 1 +Y, for * 1 {*, takin into account the indeendence of Y 1 and Y, and Result 1, is f Z (z)= z 0 * r 1 1 *r 1(r 1 ) 1(r ) yr 1 &1 e &* 1 y 1 (z&y1 ) r &1 e &* (z&y 1 ) dy 1 1 = *r 1 1 * r e&* z (r 1 &1)! (r &1)! z =* r 1 1 *r { +(&1) r 1\ (&1)r \ r1 r 0 a r &+1, r 1 ( &1)! (z&y 1 ) r &1 y r 1 &1 1 e &(* 1 &* ) y 1 dy1 a r1 &+1, r ( &1)! z &1 (* 1 &* ) &r 1 &r + z &1 (* &* 1 ) &r &r 1+ e&* e&* 1 z z=. Corollary 1. Let Y i t1(, * i ),,...,, be indeendent Gaa rando variables. Then, if (,..., ) are inteer, the.d.f. of Z=Y 1 +}}}+Y is, for z>0, f Z (z)=k i \ S i & }}} &1 =1 1 =1 1 =1 a ri & 1 +1, r 1 * i (* i &* 1 * i ) 1 & &r 1 *i a 1 & +1, r * i (* i &** i ) & 1 &r *i a & & &1 +1, r* i &1 (* i &** i &1 ( &1 &1)! ) &1 & & &r * i &1 z &1 &1+ e&* i z (for * i {* i$, i, i$#[1,..., ], i{i$) \ = *S 1(S) zs&1 e &*z for * 1 =}}}=* =* + (4) where f Z (z) is f Z (z; * 1,..., * ; r 1,..., r ), the coefficients a, r* are iven by (), K i = * i, S i=(&1) S&, S=, (5)

5 90 CARLOS A. COELHO ** i is the th eleent of the set [* 1,..., * ]"[* i ] and siilarly r* i is the th eleent of the set [r 1,..., r ]"[ ], where "'' denotes set difference. Equivalently ** i and r* i ay be defined as ** i i> = {* * +1 i r* i = { r r +1 i> i,..., ;,..., &1. Proof. Theore 1 enables us to obtain the distribution of Z 1 =Y 1 +Y. Then alyin Result 1 to the oint distribution of Z 1 and Y 3 we et the distribution of Z =Z 1 +Y 3 =Y 1 +Y +Y 3, and so on. For * 1 =}}}=* =*the result is known. K Distributions (3) and (4) ay be seen as eneralizations of the coon Gaa distribution. Indeed, if * 1 =* in (3) or * 1 =}}}=* in (4) then we have a coon Gaa distribution and if r 1 =r =1 in (3) or r 1 =}}}=r =1 in (4) we have the su of indeendent exonentials. So we ay call the distributions (3) and (4) as Generalized Inteer (GI) Gaa distributions. As a atter of fact further sile chanes in the suation order yield distribution (4) under the for f Z (z)=k i P i (z) e &* i z (z>0) (6) where P i (z) is a olynoial of deree &1 in z which ay be written as where P i (z)= c i, k (, r, * ) z k&1 (7) c i, k (, r, *)=S i 1 =k 1 =k &3 }}} & =k 1 ; }}}# & $ k (8) with r =(r 1,..., r )$ * =(* 1,..., * )$ (9)

6 GENERALIZED GAMMA DISTRIBUTION 91 and 1 =a ri & 1 +1, r* i 1 (* i &** i 1 ) 1 & &r 1 *1 ; =a 1 & +1, r* i (* i &** i ) & 1 &r *i (10) # & =a &3 & & +1, r* i (* & i&** i ) & & &3 &r * i & & $ k = a & &k+1, r* i &1 (* i &** i &1 (k&1)! )k& & &r * i &1. Exression (6) is easily obtained fro (4) by sily interchanin the suations. We ay note that, usin the notations in (10) and k for &1, distribution (4) ay be written as f Z (z)=k i \ S i 1 =1 =K i \ S i =K i { 1 = =1 \ S i 1 =k =1 &3 ; }}} &3 }}} & =1 & & =1 1 =k &3 }}} & =k & # & $ k z k&1+ e&* i z 1 ; }}}# & $ k z k&1+ e&* i z 1 ; }}}# & $ k+ zk&1= e&* i z. c i, k Exression (8) is quite easy to coute but ay becoe a bit lon even for oderately lare values of, iven the nestin of the suations. However, a little alebra will allow us to et a uch faster and easier way to coute the coefficients c i, k (, r, * )(,..., ), so that in any cases the coefficients ay even be couted by hand. Fro (8) we can see that, for a iven i, the easiest coefficient to coute is the one associated with the hihest deree in the olynoial, which ay be iven by Then, for,..., &1, we have c i, ri &k(, r, * ) = 1 k k c i, ri (, r, *)= 1 ( &1)! (* &* i ) &r. (11) {i ( &k+ &1)! R( &1, i,, r, *) c i, ri &(k&)(, r, *), (1) ( &k&1)!

7 9 CARLOS A. COELHO where R(n,,, r, *)= i{ (* &* i ) &n&1, (n=0,..., &1). (13) Usin the above notation it is then very easy to et the cuulative distribution function (c.d.f.) of Z. For non-neative inteer and real *, we obtain w 0 z e &*z dz=! * +1{ 1& \ i=0 * i w i, w>0, (14) i! + e&*w= on interatin by arts. Then, fro (6), (7) and (14), the c.d.f. of Z is F Z (z)=k i where P i *(z) is a olynoial of deree &1 in z, with P*(z)= i c i, k (, r, * ) (k&1)! * k i { P i *(z) (z>0) (15) 1& \ k&1 =0 * i z! + e&* i. (16) z= Exressions (15) and (16) show that F Z (z) ay be seen as a eneralization of the Incolete Gaa function, as one would exect. Indeed, the exression for the c.d.f. of Z ay be further silified since fro (15) and (16) F Z (z)=k i &K i c i, k (, r, *) (k&1)! * k i k&1 z c i, k (, r, *)(k&1)!! * k& i e &* i z =0 where, after soe alebraic aniulation we ay show that c i, k (, r, * ) (k&1)! * k i = * & i =(K i )&1. Thus F Z (z) ay be further written as F Z (z)=1&k i =1&K i e &* i z k&1 c i, k (, r, *)(k&1)! =0 1! z * k& i P i **(z) e &* i z (z>0) (17)

8 GENERALIZED GAMMA DISTRIBUTION 93 where P**(z)= i k&1 c i, k (, r, *)(k&1)! =0 1! z * k& i (18) is a olynoial of deree &1 in z. Now F Z (z) is clearly shown to be a eneralization of the Incolete Gaa Function. In soe cases, as we will see in Theore, the distribution of U=e &Z ay be also souht. Fro the.d.f. of Z we can readily obtain the.d.f. of U, throuh the transforation of variable Z=&lo(U). Since the Jacobian of the transforation is 1u, then f U (u)=k i P i (&lo u) u * i &1 (0<u<1) (19) where P i (&lo u), still iven by (7), is now a olynoial of deree &1 in (&lo u) with coefficients c i, k. Then the c.d.f. of U is iven by F U (u)=k i P i **(&lo u) u * i (0<u<1) (0) where P i **(&lo u), a olynoial of deree &1 in (&lo u), is iven by (18). The above exression for F U (u) ay be obtained either fro (19) and (18), usin, for non-neative inteer and real *, w 0 (&lo l) l * dl=w *+1! i=0 1 (&lo w) i i! (*+1) &i+1, 0<w<1 which is easily obtained on interatin by arts or, equivalently, throuh the chane of variable z=&lo t F U (u)= u 0 f U (t) dt= u 0 f &lo U (&lo t) 1 t dt= 0 & f &lo U (&lo t) 1 t dt u = + &lo u f Z (z) dz= + f Z (z) dz& &lo u f Z (z) dz 0 =1&F Z (&lo u). (1) Exression (1) clearly shows that it is indeed equivalent to use Z=&lo U or U=e &Z to carry out tests. For exale, usin (1) above, we see that the -ercentile of U is equal to the exonential of the syetrical of the (1&)-ercentile of Z, and conversely, the -ercentile of Z is equal to the syetrical of the loarith of the (1&)-ercentile of U. 0

9 94 CARLOS A. COELHO A articular and iortant GI Gaa distribution arises in the followin situation. Theore. Let = be an even inteer, with 1, and Y tb \a, b +,..., be indeendent rando variables with Beta distributions, where b is a ositive inteer and a =k& (,..., ), with k>. Further let and W$= Y W=&lo W$=& lo Y. Then W is the su of +b& indeendent Gaa distributions with araeters * =k+( &&1)(,..., +b&) and inteer araeters r iven by r = {h r & +h, =3,..., +b& () where h =(nuber of eleents of [, b] reater or equal to )&1. (3) Thus the.d.f. and c.d.f. of W are and +b& f W (w)=k +b& P (w) e &* w (w>0) (4) +b& F W (w)=1&k +b& P **(w) e &* w (w>0), (5) with K +b& iven by (5) and P and P ** iven by (7) and (18) resectively. The.d.f. and c.d.f. of W$ are thus +b& f W$ (w)=k +b& P (&lo w) w * &1 (0<w<1) (6)

10 GENERALIZED GAMMA DISTRIBUTION 95 and +b& F W$ (w)=k +b& P **(&lo w) w * (0<w<1). (7) Here, the coefficients c, k in the definitions of P and P ** are iven by S&r c, r (, r)= (r &1)! >, i{ (i&) (&1) r +1 S&r = (r &1)! > & (i&) > i= + (i&) (8) and, for,..., r &1, c, r &k(, r )= 1 k k where (r &k+i&1)! R(i&1,,, r) c (r &k&1)!, r &(k&i)(, r), (9) R(n, i,, r)= {i r \ i&+ n+1 (n=0,..., r &1), (30) with r as in (9). In (8), S=r 1 +}}}+r =b is the nuber of exonentials that incororate the distribution of W. Proof. Given the indeendence of the Beta rando variables, the h th oent of W$ is E(W$ h )= and thus the characteristic function of W is E(e itw )=E(e &it lo W$ )=E(W$ &it )= 1(a +h) 1(a +(b)) 1(a +(b)+h) 1(a ), (31) 1(a &it) 1(a +(b)) 1(a +(b)&it) 1(a ), where i=(&1) 1 and t is a real constant. But then, iven that = is an even inteer, usin the dulication forula for the Gaa function 1(z)=? &1 z&1 1(z) 1(z+ 1 )

11 96 CARLOS A. COELHO we ay write, and siilarly Then 1(a &it)= 1 \a + b + =? 1 \a + b &it + =? E(e itw )= = = =? 1 \k&&1 &it + 1 \ k& 1 \k& +1 &it + 1 \ k& 1(a )=? 1 &it + &it + (k& )&it&1 1(k&&it) 1 (k& )+b&1 1(k&+b) 1 (k& )+b&it&1 1(k&+b&it) 1 (k& )&1 1(k&). (k& )+b&it&1 (k& )&1 1(k&&it) 1(k&+b) (k& )&it&1 (k& )+b&1 1(k&+b&it) 1(k&) 1(k&+b)1(k&) 1(k&+b&it)1(k&&it) which, usin for all real or colex a and inteer n, ives E(e itw )= = b&1 i=0 +b& = \ 1(a+n) 1(a) n&1 = i=0 (a+i), (k&+i)(k&+i&it) &1 b&1 i=0\ k&+i +\ k&+i r k+&&1 + &it + &1 \ +&r k+&&1 &it (3)

12 GENERALIZED GAMMA DISTRIBUTION 97 where r are inteers iven by (), with h defined as 1,..., in(, b) h ={0 +in(, b),..., ax(, b), &1 +ax(, b),..., +b& or equivalently defined as in (3). Exression (3) shows that W is the su of S=b= b indeendent exonentials with araeters k& +i (,..., ; i=0,..., b) or the su of +b&=+b& indeendent Gaa rando variables with araeters r and * =k+( &&1) (,..., +b&). This way the.d.f. of W is a GI Gaa distribution, as defined in (4), with = +b& * =k+ &&1. Thus, in order to obtain the.d.f. and c.d.f. of W in (4) and (5) we ust have to relace by +b& and Z by W in (6) and (17). Siilarly, the.d.f. and c.d.f. of W$ in (6) and (7) are obtained by relacin by +b& and U by W$ in (19) and (0). The coefficients c, k,,..., r ;,...,, in the olynoials P and P** are iven by (11) throuh (13), which in this articular case, iven that * i &* =(i&), ay be written as in (8) throuh (30). K We ay notice that in Theore, and b are interchaneable as lon as they are both even, and that when b=1 the distribution in (6) becoes a Beta distribution. Theore ay be seen as an extension of the well known result &n lo[x(x+y)]t/ (Fuikoshi and Mukaihata [5]), for two indeendent rando variables Xt/ n and Yt/. 4. APPLICATION TO THE EXACT DISTRIBUTION OF THE (GENERALIZED) WILKS' 4 An useful and interestin alication of the GI Gaa distribution is in obtainin the distribution of soe ultivariate statistics. The Wilks' 4 statistic is aon such statistics.

13 98 CARLOS A. COELHO The Wilks' Labda (Wilks [11, 1]) is a well known statistic used, under a ultivariate noral settin, to test in ultivariate analysis of variance the existence of overall differences aon the level eans of a factor, in ultivariate reression or canonical analysis to test the equality of the vector of reression araeters or art of it to a iven vector or still used to test the indeendence of sets of norally distributed variables (Wilks [11, 1]; Bartlett []; Anderson [1, Cha. 8, 9]; Kshirsaar [8, Cha. 8]). Since all the above settins lead to the sae test statistic, in order to kee it short we will describe this statistic only under the last settin. Let x be a _1 vector of variables with a oint -ultivariate noral distribution N (+, 7). Further let x be slit into subvectors, the kth of which has k variables, with = 1 +}}}+. Then each subvector xk(,..., ) has a k -ultivariate noral distribution N k (+ k, 7 kk ) and, for a sale of size n, the (n)th ower of the likelihood ratio statistic to test the null hyothesis H 0 7=dia(7 11,..., 7 kk,..., 7 ), (33) i.e., the hyothesis of indeendence of the subvectors x k, is the Wilks' Labda, where } stands for the deterinant and 4= A > A kk, (34) A 11 }}} A 1k }}} A 1 b b b A=_A k1 }}} A kk }}} A b b b A 1 }}} A k }}} A k& is either the sale variance-covariance atrix of the variables in x or the Maxiu Likelihood Estiator of 7. For > such a test statistic also arises under the settin of the Generalized Canonical Analysis, when the aroach roosed by Carroll [3] is considered (Coelho, [4, Cha. 4]). A eneral concise exression for the exact distribution of the (eneralized) Wilks' Labda, without involvin any series exansion or unknown coefficients, ay then be obtained by direct alication of the result in Theore as lon as at ost one of the sets of variables has an odd nuber of variables.

14 GENERALIZED GAMMA DISTRIBUTION The General Case of Sets of Variables For eneral the Wilks' 4 statistic in (34) to test the indeendence of the sets of variables x1,..., x, ay be written as &1 4= 4 k(k+1,..., ) (35) where 4 k(k+1,..., ) stands for the Wilks' 4 statistic to test the indeendence of the set of variables xk and the set fored by the oinin of the sets (x k+1,..., x) (Anderson [1, Theore 9.3.]; Coelho [4, Sec. 4.7]). Under the hyothesis of oint ultivariate norality of the sets of variables and the null hyothesis (33), of indeendence of the sets of variables, the &1 Wilks' Labda statistics 4 k(k+1,..., ) are all indeendent (Coelho [4, Cha. 4]). Considerin that xk has k variables (,..., ), the distribution of 4 k(k+1,..., ) is the sae as the distribution of > k Y where, for a sale of size n+1, with n 1 +}}}+,Y are k indeendent Beta rando variables with Y tb((n+1&q k &), q k ), usin q k = k+1 +}}}+ (Anderson [1, Theore 9.3.]). Then we have the followin Theore. Theore 3. For the eneral case of sets of variables, the kth set havin k variables (,..., ), when at ost one of the sets has an odd nuber of variables, and for a sale of size n+1> 1 +}}}+, the.d.f. and c.d.f. of W=&lo 4, under the null hyothesis (33) are iven by and resectively, with f W (w)=k F W (w)=1&k P (w) e &* w P **(w) e &* w, = & where = k, (36) bein K iven by (5) and P (w) and P ** defined as in (7) and (18). The.d.f. and c.d.f. of 4 are then iven by f 4 (l)=k P (&lo l) l * &1

15 100 CARLOS A. COELHO and F 4 (l)=k P **(&lo l) l *. The scale araeters are * =(n&+ ) (,..., ), with iven by (36) above. The shae araeters r (,..., ), are iven by r = {h r & +h, =3,..., = 1 +}}}+ &, an extension of (), with the h 's defined as h =(nuber of k (,..., ) )&1,,..., = &. Proof. have Given the indeendence of the &1 Wilks' Labdas in (35) we &1 E(4 h )= E(4 h k(k+1,..., ) ). When at ost one of the sets has an odd nuber of variables, without any loss of enerality, we ay suose it to be the th set. Then, fro the roof of Theore in Section 3, &1 E(e &it lo 4 )=E(4 &it )= k +q k & \ _ k&q k + +&r k \n& &it & = \ n&+ r + \ n&+ n& r k&q k + k + &it +&r where q k = k+1 +}}}+, and r k (,..., &1;,..., k +q k &) are defined by r k = {h k r k, & +h k, =3,..., k +q k & (37) with h k =(nuber of eleents in [ k, q k ] reater or equal to )&1 (38)

16 GENERALIZED GAMMA DISTRIBUTION 101 and &1 r = r k (39) with r k =0 if > k +q k &. Therefore r ay be defined by (), where now &1 h = h k =(nuber of k (,..., ) )&1,,..., &. (40) This shows that the distribution of &lo 4 is, for, the su of & Gaa distributions with scale araeters * =(n&+ )(,..., &) and shae araeters r. Thus alyin Theore we obtain the results in this Theore. K We ay note that for = we obtain the distribution of the usual Wilks' Labda and thus of each of the &1 Wilks' Labdas 4 k(k+1,..., ) (,..., &1) on the riht hand side of (35). Further, we ay also note that the shae araeters r, in the distribution of the eneralized Wilks' Labda, are the ordered su of the shae araeters in the distribution of the &1 Wilks' Labdas 4 k(k+1,..., ) (,..., &1). For exale, for =3, with 1 =4, =4 and 3 =3, we have = &=9, and fro (35) throuh (38), h =[h ]=[,,,1,&1,&1,&1,&1,&1]$ r =[r ]=[,,4,3,3,,,1,1]$ r 1=[r 1 ]=[1,1,,,,,,1,1]$ r =[r ]=[1,1,,1,1]$, where the eleents of r 1 are the shae araeters in the distribution of 4 1(, 3), the Wilks' Labda statistic used to test the indeendence between the first set and the suerset fored by oinin the second and third sets of variables, while in r are the shae araeters in the distribution of 4 (3), the Wilks' Labda statistic used to test the indeendence between the second and third sets of variables. 5. DISCUSSION The GI Gaa distribution and the distributions resented in Theore of Section 3 are of aor relevance in obtainin the distribution of several ultivariate statistics whose oents are of the for in (31). Based on the

17 10 CARLOS A. COELHO above results it is ossible to obtain the distributions of these statistics in a eneral, ore concise and anaeable for. This will ake easier the coutation of ercentiles and enable us to overcoe the robles arisin fro the use of distributions under a series exansion for. As an illustration, the null distribution of the Wilks' 4 statistic to test the indeendence of sets of variables is exlicitly obtained in a sile anaeable for even for eneral. Moreover, the use of the GI Gaa distribution also enables us to et a deeensiht and at the sae tie have an overall view uon the studies carried out so far on the distributions of a nuber of ultivariate statistics, naely the Wilks' Labda. ACKNOWLEDGMENTS The author thanks and exresses his earnest areciation for the insihtful coents fro the editor and a referee. REFERENCES 1. Anderson, T. W. (1958). An Introduction to Multivariate Statistical Analysis. Wiley, New York.. Bartlett, M. S. (1938). Further asects of the theory of ultile reression. Proc. Cabride Philos. Soc Carroll, J. D. (1968). Generalization of canonical correlation analysis to three or ore sets of variables. In Proc., 76th Annual Convention of the Aerican Psycholoical Association, 1968, Coelho, C. A. (199). Generalized Canonical Analysis, Ph.D. thesis. The University of Michian, Ann Arbor, MI. 5. Fuikoshi, Y., and Mukaihata, S. (1993). Aroxiations for the quantiles of student's t and F distribution and their error bound. Hiroshia Math. J Guta, R. D., and Richards, D. (1979). Exact distributions of Wilks' 4 under the null and non-null (linear) hyotheses. Statistica Kabe, D. G. (196). On the exact distribution of a class of ultivariate test criteria. Ann. Math. Statist Kshirsaar, A. M. (197). Multivariate Analysis. Dekker, New York. 9. Nandi, S. B. (1977). The exact null distribution of Wilks' criterion. Sankhya~ Tretter, M. J., and Walster, G. W. (1975). Central and noncentral distributions of Wilks' statistic in Manova as ixtures of incolete beta functions. Ann. Statist Wilks, S. S. (193). Certain eneralizations in the analysis of variance. Bioetrika Wilks, S. S. (1935). On the indeendence of k sets of norally distributed statistical variables. Econoetrika Printed in Beliu