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ON THE MARGINAL DISTRIBUTIONS OF THE LATENT
ROOTS OF THE MULTIVARIATE BETA MATRIX
By
A. W. Davis
to
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 690
July 1970
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ABSTRACT
On the marginal distributions of the latent roots of
the multivariate beta matrix
The marginal distributions of the latent roots of the multivariate beta
matrix are shown to constitute a complete system of solutions of an ordinary
differential equation (d.e.), which is related to the author's d.e. 's for
Rotelling's generalized T2 and Pillai's V(m) statistics.
o
Similar results are
given for the latent roots of the multivariate F and Wishart matrices (E=I).
Pillai's approximations to the distributions of the largest and smallest roots
are interpreted as exact solutions, the contributions of higher order solutions
being neglected.
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On the marginal distributions of the latent roots
of the multivariate beta mattix*
A. W. Davis
C.S.I.R.O., Adelaide, and University of North Carolina
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1.
Introduction.
Let S(mXm) and T(mxm) have independent Wishart distributions
W(q,E) and W(n,E), respectively, where E is the population covariance matrix and
q,n
~
m.
The latent roots £1 > ••• > £m > 0 of the multivariate beta matrix
B = S(S+T)-l are well known to have the joint density function
(1)
m ~(q-m-l) m
~(n-m-l)
TI'l(l-£,)
TI,<,(£,-£,),
<l>m;q,n(&) = K(m;q,n)IT,~= l£i
~=
~
~ J
~
J
(2)
K(m;q,n)
= ~~~:~[r(~(q+n-i»/r(~(m~i»r(~(q-i»r(~(n-i»],
The marginal distributions of the individual £, have been investigated by
~
.
Roy [14], [15], who showed that the largest root £1 is of basic importance in
testing hypotheses and constructing confidence regions in multivariate analysis
of va~iance; also by Pillai [10], Khatri [8], Sugiyama and Fukutomi [17],
Sugiyama [16], and AI-Ani [1].
Pillai [11] gave very accurate approximations
to the upper and lower tails of the distributions of £1 and £m' respectively,
and £1 has been extensively tabulated by Heck [7] for m<5, using Pillai's
approximation, and Pillai ([11], [12], etc.) for m<20.
Studies of the noo-
central distributions have been made by Khatri [9] and Pillai and Dotson [13].
*This research was· supported by the National Institutes of Health Grant No. GM-12868.
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2
As
n~,
matrix.
nB~WI'
say, having the distribution W(q,I) where I(rnxm) is the unit
Ranumara and Thompson [6] have tabulated the largest and smallest roots
of WI using limiting forms of Pi11ai's approximations, and discussed their
application.
The present author [3], [5] has shown that the null distributions of tr B
(Pi11ai's V(m»
and tr F (Rote11ing's genera1izedT 2 ), where F = ST- 1 , satisfy
o
certain ordinary linear differential equations (d.e. 's) of order m which are
related by a simple transformation.
distributions of the 2
i
In Section 2 it is shown that the marginal
form a complete system of solutions of a similar d.e.
Thus, the power-series of Sugiyama and Fukutomi are solutions at the regular
singularities 0 and 1.
Pi11ai's approximations are also shown in Section 4 to
be exact solutions of the d.e., but approximations to the distributions insofar
as contributions from higher-order solutions' are neglected.
Corresponding results
for the latent roots of F and WI are readily deduced (Sections 6 and 7).
2.
The differential eguation.
r
CR , where R is the real line.
{O<x <••• <x <2<x
.
r
,8
s- 1< .•• <x 1<1}
The marginal density function f (2) of
s
given by
(3)
f
s
(2) =
fDm- 1 (s,2) ~m;q,n (x1 '···,xs- l'~'x8 , ••• ,xml)dx
--
~1
-
where dx = IT =l dx ; it is proportional to
i
i
~~(q-m-1)(1_~)~(n-m-1)f
. ~(x)~-l(~-x.)dx
~-'
. Dm-1 (s,~) - ~=1
(4)
in which
~
denotes
~
m-1;q-1,n-1'
Define
~
s
is
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3
(5)
'1' (J/,;x) = <1>(x)I (J/,-x (l»· •• (J/,-x (
r
-
a
a m- 1-r
a
»,
(r=O,l, ••• ,m-l),
the summation being extended over the (m-l) selections of integers 0'.(1) < ••• < O'.(m-l-r)
r
from the set 1,2, ••• ,m-l.
When r=m-l, the sum is taken to be unity.
We now
introduce the mfunctions
L
(J/,) = f m. 1
'1' (J/,;x)dx, (r=O,l, ••• ,m-l),
s,r
D - (s,J/,) r
- -
(6)
noting that f (J/,) is proportional to J/,~(q-m-l)(l_J/,)~(n-m-l)L
s
s,O
Our object is
to show that, for each s, the L
are related by a system of first-order differential
s,r
equations which are independent of s.
Differentiating (6),
L' (J/,) = - Z(l) + Z(2) + (r+l)L
+1'
s,r
s,r
s,r
s,r
(7)
where
Z(l)
s,r
=
f Dm- 2 (s-l,J/,) '1'r (J/,;xl,···,x s- 2'J/"x s- l'···'xm2)dx,
-.-
(8)
Z(2) =
f Dm- 2 (s,J/,) '1'r (J/,;xl,···,x s- l'J/"x s , ••• ,xm2)dx.
s,r
-Now let a(l), ••• ,S(r) denote the set of subscripts complementary to
a(l), ••• ,O'.(m-l-r).
We have
rJ/, L
s,r
(9)
= (m-r)L
. s,r+1 + 0 s,r ,
say.
Integration by parts with respect to the xS(i) yields
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~(q+n-2m+2)G
(10)
s,r
= t(1-t)[Z(1)_Z(2)]
s,r s,r
+ ~(q-m+r)L
s,r
+~
s,r
,
where
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~
s,r
(11)
A term similar to (11) occurred in the derivation of a d.e. for Rote11ing's
generalized To2 ([3], Equation (2.13», and the same approach yields
~
(12)
s,r
= ~(m-r)(m+r-3)L
s,r- 1+l~(r-1)(1-2t)L s,r +l~(r+1)t(1-t)L s,r+1.
Finally, eliminating the Z's, G's and
~'s
from equations (7), (9), (10) and (12),
we find that
t(l-t)L'
=
s,r
~(m-r)(q+n-m+r-1)L
+
~(r+1)(r+2)t(1-t)L
s,x-
l+l~r[(l-t)(q-m+r)-t(n-m+r)]L
s,r
( 13)
where Ls,- 1
s,r+1' (r=0,1, ••• ,m-1),
=Ls,m =O.
We observe that the system (13) is independent of s, and in principle one
could successively eliminate L 1, ••• ,L
l' arriving at a homogeneous linear
s,
s,md.e. of order m having each L 0 as a solution.
s,
Clearly the f
s
will be solutions
of a similar d.e.; furthermore, they will constitute a linearly independent and
hence complete system of solutions, since as
(14) .
where
t~O+
fS(t)/t~(q-m-1) ~ ks(m;q,n)t~(m-s)(q-S+2), (s=l, ••• ,m),
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k (m;q,n) = K(m;q,n)/[K(s-1;q+m-s+l,n-m+s-l)K(m-s;q-s,m-s+3)].
(15)
s
This is easily proved by writting x. =
.
J
(j=s, ••• ,m-l) in (4) and letting
~Wj
~+o.
We note in addition that (13) is invariant under
(16)
q+n, n+q,
~+1-~,
r
provided that we also replace L,
by (-1) Ls,r; this reflects the obvious result
. s,r
that (16) transforms fs(~) into fm+l-s(~).
3.
Solutions of the d.e.
It is convenient to introduce H =
r
.
(l-~)
-r
L
s,r
(r=O,1, ••• ,m-1) and to express (13) as a matrix d.e. for H = (H , ••• ,H _ )':
O
m1
(17)
" .
where
The d.e. (17) is of Fuchsian type, with regular singularities at
~=O,l
and
infinity, and we refer to [2], Chapter 4, for the general theory of such d.e.'s.
Assuming a series solution H =
Lr : o
hr~p+r in 1~1<1, we obtain Ah O .= phO' so that
p must be one of the latent roots a , ••• ,a _ of.A, and h the corresponding
O
m1
O
•
il
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1
il·
il
il.
,
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latent vector.
To relate this fundamental set of solutions to the f (t), we
s
first obtain a non-singular transformation H=PM, where P(mxm) is independent of
t, such that P- 1AP = diag(a ).
A suitable choice is
r
(19)
P = {pij },
P ij
= (-1)
i-j m-j-1 i-I
(i.)IT . (q+n-m+r)/(q-m+j+r+l) ,
(20)
Both P and P
-1
are lower triangular, and MO=H '
O
p- 1 CP = G =
(21)
v
A ,
1
III
o
0
0
v
1
Am-1
m-2
llm-l
where
A. = (m-i) (n-i) (q+i-1) (q-m+i-1) (q-m+i-2) (q+n-m+i-1)/[2(q-m+2i-2)
~
(q-m+2i-1) 2 (q-m+2i)],
(22)
~I
ll.
~
,
il
11
llO'
It may be shown that
'v
il
il.
11j
r=J
with inverse
;1
!I
-J
= i2-~i(m+n-3)-m+1
+ ~i(i+1)(m-i)(n-i)/(q-m+2i-l)
-~(i+1)(i+2)(m-i-1)(n-i-1)/(q-m+2i+1),
v. = ~(i+1)(i+2).
~
The d.e. (17) now takes the form
,
]1
(23)
dM/dt
=
[t- 1 diag(a ) + (l-t)-l G]M,
r
~I
)1
a +r
corresponding to the latent root a
and assuming a solution M = lr:o nrt p
p
,
(no ,r , ••• ,n~,r
-1 ,) of the nr :
1
'
~I·
1
il
of A, we obtain the following recurrence relations for the components
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np,O = 1,
(24)
ni,O = 0 (i~p),
(r-a.+a
1
l+[ll.+(r-l)-a.+a
1
P)n i ,r = A.n.
1 1- ,r1
1
P]n i ,r- 'l+v.n,·+l
1 1
,r-l'
(i=O, ••• ,m-l; r=1,2, •••• ).
This form of solution unfortunately breaks down if ai-a
integer for some i.
In fact, a +l-a
p
p
=
~(q-m+l)+p
if q-m is odd, while a + -a = q-m+2(p+l) (p
p 2 p
~
p
is a positive
(p < m-2) , which is an integer
-
m-3) is always an integer.
Generally in such situations the solution must be obtained by limiting procedures
which may produce logarithmic terms.
However, it may be seen from (6) that the
Ls,r are in fact representable by power series, and it appears that if a.-a
is
1
p
a positive integer for i>p, then the right-hand side of the ith equation in (24)
vanishe~
identically when r = a.-a.
1
p
Thus n
i,r
is an undetermined constant
introducing the a.-solution at this stage, and the power series form is preserved.
1
We also see from (24) that the nO
,r
are zero for r<p in the a -solution,
p
'while
(25)
= (p+l)!/ni~l(q-m+p+i+l) = i; p , say.
a+p
-Hence M (!) = O(! p ) as !+O+, and since a
+(m-s) = ~(m-s)(q-s+2), it follows
O
m-s
from (14) that L ,s must be some linear combination of the am-s ,am-s +1'.'.' and
O
a
m-
I-solutions.
The coefficient of the a
m-s
-solution is clearly k (m;q,n)/i;
,
s
m-s
but the remaining coefficients have not been determined for general s.
However,
the density function f (!) of the largest root corresponds to the largest root
l
am-I of A, and is thus completely specified by (24).
The resulting power series
coincides with the result of Sugiyama and Fukutomi [17].
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4.
Pillai's approximations.
A particular solution corresponding to the smallest
root a =0 of A may be given explicitly as an (m-l)th degree polynomial.
o
(Z)i = z(z+l) ••• (z+i-l),(z) . = z(z-l) ••• (z-i+l), it may be shown that
-~
(26)
Thus we obtain the following approximation to the lower tail of f (t) for
m
large q by neglecting the al, ••• ,a _ solutions (i.e., terms of order t q- m+l
ml
at least):
f (t) ~ k(m;q,n)t~(q-m-l)(l_t)~(n-m-l)
m
(27)
tm-I(m-l) ( +n) ( 1)
( t)r
lr=O r
q -m r q- -(m-l-r) ,
••I
where
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large n (near the regular singularity t=l) is obtained.
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Writing
k(m;q,n) =
(28)
TI~r(~(q+n-m+l»/[2m-lr(~)r(~q)r(~)].
Using (16), a corresponding approximation to the upper tail of fl(t) for
The result is found
to be simply the right-hand side of (27) multiplied by (-1)
m-l
•
The integrated form of the approximation was arrived at by Pillai [11]
using a different approach, and used in a series of tabulations of the upper
5% and 1% points of
when n 2
~
~l.
Its accuracy to essentially five places of decimals
m+ll was demonstrated at least for m ~ 10 by substituting inexplicit
expressions for the distribution function [10].
In order to investigate the
usefulness of the d.e. (23), some percentage points were calculated by following
the a
m- I-solution .>ut from. the origin, using the same computation procedure as
in [4].
The method,tPI'<red to be effective at least up to m=7, since on
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comparing the 1% points, i.e., the less ,accurate results of the d.e. and the
more accurate results of the approximation, these were generally found to differ
by no more than a unit in the fifth decimal place.
On the other hand, the 5%
points obtained from the d.e. tended to exceed Pillai's by about three units
in the fifth decimal place.
The d.e. approach should be more accurate at lower
significance levels, and a tabulation of upper 10% points has been made.
5.
The median root for m=3.
The success of the Pillai approximation suggests
a similar approach to the tails of the distributions of the other roots, approximating the lower tail of f
s
by the a
m-s
-solution for large q, and deducing a
corresponding result' for the upper tail of f
m-s
+1 when n is large using (16).
However, so far it has proved difficult to obtain the solutions in closed forms
analogous to (27), except in the case of the median root 2
2
when m=3.
The
aI-solution is
1
1
1
M = (~q+2[2(n-l)/(q-2)-~(q+n-4)])2~q- (1-2)~,
l
(29)
kq
!m
M = -[(n-2) (q-3) (q+n-2)/2q(q-l)]2 2 (1-2)2 •
2
Hence, for large q and 2 near zero,
(30)
the beta density with parameters q-l and n-l.
the upper tail and large n, and
of 2 ,
2
ha~been
The same approximation holds for
used to compute upper 5% and 1% points
The results are identical to five decimal places with those published
by Pillai and Dotson [13], except where the latter have employed extrapolation,
in which case differences of up to 0.006 occur for the 5% points and up to .003
for the 1% points.
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6.
The latent roots of the multivariate F matrix.
Substituting t = v/(v+l) in
(23), we obtain the d.e.
l
) + (l+v)-l [G-diag(a )])M*,
dM*/dv = (v- diag(ar
r
(31)
which has regular singularities at 0, -1 and infinity.
f*(v) of the sth largest root v
s
a
m-s
-
of F is a certain linear combination of the
, ••. ,a I-solutions of (31), multiplied by the factor v~(q-m-l)/(v+l)~(q+n)-m+l.
m-
a
+(m-s)
f*(v)/v~(q-m-l) - k (m·q n) v m-s
.
(32)
s
where k
7.
s
The density function
s
s
"
is defined by (14).
As
The latent roots of the Wishart matrix WI.
and nv
s
n~,
the densities of nt
converge to f (u), say, the density function of the sth largest root
s
To obtain a d.e. for the f , we substitute t = u/n and M.
s
n~
(i=0, ••• ,m-1) in (23); 1eting
~
it is found that
1
dM/du = [u- diag(a
. r) +
(33)
where M = (M ' ••• ,M _ )', and
O
ml
(34)
G=
110
-
V
~
II
-
0
o
.
VI •
·V
m-2
0
where
s
~
m-l
-
llm-1
G]
M,
= n
i -
M.
~
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(35)
Xi
= (m-i) (q+i-1) (q-m+i-1) (q-m+i-2)/[2(q-m+2i-2) (q-m+2i-1) 2 (q-m+2i)]
-
= -[i(q-m+i) (q+m-5)+2(m-1)(q-m-1)]/[2(q-m+2i-1) (q-m+2i+1) ].
11 i
The d.e. (33) has a regular singularity at u=O, but an irregular singularity
at infinity.
As before, f
s
is a linear combination of the a
at the origin, multiplied by e -~ u ~(q-m-1) •
, ••• ,a 1-solutions
m-
In particular, as u+O+,
a
-f (u)/u~(q-m-1) - k- (m;q ) u m-s
s·
.s
(36)
m-s
+(m-s)
where
ks (m;q) = K(m;q)/[K(s-1;q+m-s+1)K(m-s;q-s,m-s+3)]
(37)
K(m;q) =
~~/[2~q ~:~ r(~(m-i»r(~(q-i»] •
Taking limits in (27), we find that f (u) ~ (_1)m-1 gm,q (u) for large u,
1
and f (u) ~ g
(u) for u near zero, where
m
m,q
. gm,q(u) = K(m;q)e
-~ ~(q-m-1)
u
~m-1
m-1
r
lr=O( r )(q-1)_(m_1_r)(-u) ,
(38)
Again, g
corresponds to a particular a -so1ution of (33) at the origin.
O
m,q
Banumara and Thompson [6] used integrated forms of (38) in their tabulation of
u •
m
Some percentage points of u
1
were calculated using the a _ -so1ution
m1
of the d.e., and these differed from the results of the approximation by no more
than a unit in the second decimal place.
The d.e.(33} is in fact closely connected with the author's d.e. [5] for
the moment generating function'of Pi11ai's V(m) = tr B.
If we write
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A
(s) =lexp (-s v(m», then it is easily seen that fl(u) is proportional
m;q,n
to e
-~
u
~q-l
A
l l ,m+2(~)'
m- ;q-
The author's thanks are due to Mrs. Darien Wei and Dr. R. W. Helms of
the Department of Biostatistics, School of Public Health, University of
North Carolina, for their assistance with the computing.
REFERENCES
[1]
AI-Ani, S. (1968). On the distribution of the ith latent root under null
hypotheses concerning complex multivariate normal populations. Mimeo
Series 145, Department of Statistics, Purdue University.
[2]
Coddington, E. A. and Levinson, N. (1955).
Equations. McGraw-Hill, New York •
[3]
Davis, A. W. (1968). A system of linear differential equations for the
distribution of Hotelling's generalized T2 • Ann. Math. Statist. 39, 815-832.
Theory of Ordinary Differential
o
[4]
Davis, A. W. (1970). Exact distributions of Hotelling's generalized To2 •
Biometrika 57, 187-191.
[5]
Davis, A. W. (1970). On the null distribution of the sum of the roots of
a multivariate beta distribution. To appear in Ann. Math. Statist.
[6]
Hanumara, R. C. and Thompson, W. A. (1968). Percentage points of .the
extreme roots of a Wishart matrix. Biometrika 55, 505-512.
[7]
Heck, D. L. (1960). Charts of some upper percentage points of the distribution of the largest characteristic root. Ann. Math Statist., 31, 625-642.
[8]
Khatri, C. G. (1964). Distribution of the largest or the smallest
characteristic root under null hypothesis concerning complex multivariate
normal populations. Ann. Math. Statist. 35, 1807-1810.
[9]
Khatri, C. G. (1969). Non-central distributions of ith largest characteristic roots of three matrices concerning complex multivariate normal
populations. Ann. Inst. Statist. Math. 21, 23-32.
[10]
Pillai, K. C. S. (1954). On some distribution problems in multivariate
analysis. Mimeo Series 88, Institute of Statistics, University of
North Carolina.
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[11]
Pi11ai, K. C. S. (1965). On the distribution of the largest characteristic
root of a matrix in multivariate analysis. Biometrika 52, 405-414.
[12]Pi11ai, K. C. S. (1967). Upper percentage points of the largest root of
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