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On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix
JOURNAL
OF MULTIVARIATE
4,265-282
ANALYSIS
(1974)
On the Evaluation of Some Distributions that Arise Simultaneous Tests for the Equality of the Latent Roots of the Covariance Matrix P. R. Aerospace
Research
KRISHNAIAH Laboratories,
AND
F. J.
Wright-Patterson
in
SCHUURMANN* Air
Force
Base,
Ohio
45433
In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given.
1.
INTRODUCTION
There are several physical situations where the experimenter is interested in testing for the equality of the latent roots of the covariance matrices of real and complex multivarite normal populations. Krishnaiah and Waikar [15, 161 proposed certain procedures for testing the hypothesis of the equality of the latent roots of the covariance matrix of the real multivariate normal population. Similar procedures can be used in the complex case also. These procedures are also useful in drawing inference on certain subhypotheses when the total hypothesis is rejected. The distributions of the ratios of the extreme roots and the ratios of the individual roots to the sum of the roots of the real and complex Wishart matrices arise in the procedures proposed in [15, 161 and some other test procedures discussed in this paper. The purpose of this paper is to investigate the evaluation of the probability integrals of the above distributions and give percentage points which are useful in the application of the above procedures. AMS 1970 subject classifications: Primary 62HlO; Secondary 62E15. Key words and phrases: Exact distributions, ratios of the roots, Wishart matrix, multiple time series, tests for additivity, simultaneous tests, mixture of populations. * The work of this author was done at the Aerospace Research Laboratories while in the capacity of a Technology Incorporated Visiting Research Associate under Contract F33615-71-C-1463. This work may be reproduced in part or whole for any purpose of the United States Government. The present address of this author is Miami University, Oxford, Ohio.
265 Copyright All rights
0 1974 by Academic Press, Inc. of reproduction in any form reserved.
266
KRISHNAIAH
AND
SCHUURMANN
In the case of the real Wishart matrix, Davis [5] established a useful relationship between the Laplace transformations of the ratios of the individual roots to the sum of the roots and the densities of the above individual roots. Using this relationship, and the expressions in [1 1] for the densities of the extreme roots of the real Wishart matrix, Schuurmann, Krishnaiah, and Chattopadhyay [20] derived expressions for the distributions of the ratios of the extreme roots to the trace of the real Wishart matrix; they have also constructed tables for these distributions. In Section 2 of this paper, the authors derive expressions for the distributions of the ratios of the intermediate roots to the trace of the real Wishart matrix by using Eq. (2.1) of [5] and the expressions given in Krishna& and Waikar [14] for the intermediate roots of the real Wishart matrix. Tables for these distributions are also constructed for some values of the parameters. In Section 3, the authors obtain a relationship between the Laplace transformations of the ratios of the individual roots to the trace of the complex Wishart matrix and the distributions of the individual roots of this matrix. Using this relationship and the expressions given in Krishnaiah [IO] for the densities of the individual roots of the complex Wishart matrix, the authors obtain expressions for the distributions of the ratios of the individual roots to the trace of the above matrix. They have also constructed tables for these distributions. The evaluation of the probability integrals connected with the ratios of the extreme roots of the real and complex Wishart matrices is discussed in Section 4. Tables for the percentage points of these distributions are given in Section 5. Finally, the applications of the distributions considered in this paper are discussed in Section 6.
2. DISTRIBUTIONS
OF THE RATIOS OF THE INTERMEDIATE ROOTS TO THE TRACE OF THE REAL WISHART MATRIX
Let Zi < **. < 1, be the latent roots of the real Wishart matrix S with n degrees of freedom and E(S) = nI, , where I, is the identity matrix of order p x p. Then, the joint density of the above roots is well known [I93 to be
where
0 < Zr < ... < Z, < co, r = (n - p -
k(p,y)=
,+P(1/2)9(2*+~+1)/2 p VW
1)/2 and
+ P + 2 - W) WJ + 1 - i)/2)].
LATENT
ROOTS
OF THE
COVARIANCE
267
MATRIX
In this section, we derive the densities of ua ,..., u,-r where ui = l&Z1 lj for i = l,...,p. Let f,‘fi and gr’,, denote the densities of uj and 1, , respectively. Davis [S] showed that ,Ee{(l + w)(Zrp+p”+p-4)/2~~~(1/(1 = 2T(p(2r
+ w))}
+ p + 1)/2) e’s’2-p(zz+P+1))/2gai~(2F),
(2.2)
where Y(h(w)) denotes the Laplace transformation of h(w). Krishnaiah Waikar [14] showed that the density of li (2 < j < p - 1) is given by gg.(w)
and
= k( p, t) Cl x2 (- 1 )v+~~+~::~:%Jl(W)W”’ x P(&j-
l,{k,
,-e-s k,-d,o,
wu)&;
P-
j,&
,..., Bp-j), W, 4,
(2.3)
where #(w) = exp(-w/2) wr and v = )( jT + j + 2). Also, (K, < ... < R,-l} is a subset of the set of integers (0, l,...,p - l}, {tr < ... < t,-i+l} is the over subset complementary to {k, < ... < k,-,), and zr denotes the summation all (if,) possible choices of {k, < ... < kjpl}. Similarly, CY~is a subset (with one element) of the set of integers {tl ,..., t,-j+l}, {& < ... < j3p--i} is the subset complementary to q , and x2 denotes the summation over all (p - j + 1) possible choices for .q . Also, the p’s are as defined below ~(4; 2~ (4 ,..., k,,), L, U) = 44;
24 (4 >..., k,th L, U)
(2.4)
and ,4k
2 + 1, (4 ,..., k2t+J,L,
U)
2t+1 =
zl
(-l)“+‘F,,(L,
U)
GiO,k
2 + 1, (4 ,..., kzt+J,L,
u)
(2.5)
where 44;
24 (k, 9.0., k,tLL, W
= I(f:@, Uhi=1.....2t11’2,
G,(lcI;2t + 1,{k, ,..., ht+JsL u) = I(fS$L, u)h=l..... cI-l.q+l,..., St+111’2, F,(L, U) = pwds and
Making
the transformations
gi = ZJZj , (i = l,..., j -
I,j
+ I,..., p) and Zj = Zj
KRISHNAIAH AND SCHUURMANN
268
in Eq. (2.1) and integrating out g, ,..., gj-r , gj+r ,..., g, by applying Lemma 2.1 of [14], we obtain the following alternative expression for the density of 4. g:,:(w) = k(p,
Y)
exp(-wj2)
xC(-1)
~(p(~~+J+l)-~)/~
(j(i+2)/2)+~:=:acf(~l;j - 1, {al ,..., aj-l}, 0, 1)
x ~(4~; P -j,
& ,..., hj>, 1, ~0)
(2.6)
where {a, < ... < aj-r} is a subset of (0, l,...,p - 2) and {S, < ... < SPPj} is the subset complementary to {a, ,,.., a&, and C denotes the summation over all ( ;I:) possible choices of (a, < ..* < aiPl}. Also &(x) = exp(-wx/2)(
I - X) ~(P(~~+p+l)-~)/~
and #2(~) = exp(-wx/2)(x - 1) ~(P(~~+p+l)-~)/~.Starting from Eq. (2.3) (or Eq. (2.6)), we observe that g:.\(w) is of the following form gi!T(w) = k( p,
Y)
exp( -w/2) w(‘(~~+‘+~)-~)‘~gl
dij exp( -Tijw)/wmij,
(2.7)
when Y is an integer. Now using Eqs. (2.2) and (2.7), we obtain after inversion the following: f$(z)
= r((p(2Y
+ p + 1) -
2)/2) k(p, Y)2(P(2T+P+1)‘*) (2.8)
where (x)+ is equal to x or 0, accordingly, as x > 0 or x < 0. It is tedious to obtain algebraic expressions for the constants nij , dii and mij . But for fixed values of p and Y (p not very large) they can be computed using a computer. When Y is not an integer,fF,\(w) is of the same form as Eq. (2.8) with N replaced by co. 3. DISTRIBUTIONS OF THE RATIOS OF THE INDIVIDUAL
ROOTS TO THE TRACE OF THE COMPLEX WISHART MATRIX
Consider a class of random matrices whose eigenvalues cr ,..., cg have the joint distribution
fib ,***14 = K(P, a, 8,d) fi [cP exp(-dc,)] n (ci - ci)s i=l
i>j
(3.1)
LATENT
ROOTS
OF
THE
COVARIANCE
269
MATRIX
where0 < ci < ... < c, < co and K(p, a, /3, d) is a normalizing constant. Now, let oi = ci/CLl cj for i = 1, 2,..., p. Also, let h& and yF,L denote the densities of vui and cj , respectively. The joint density of va ,..., v, is seen to be
f&2
9 573 ,..*>
vP)
=
where CL, vi = 1. Starting Davis [5], we obtain
K(
P> a9
from
B9 d)(
l/d)"r(S)
fil
via
Eq. (3.2) and following
n
i>j
(Vi
-
Vj)”
(3.2)
the same lines as in
(3.3)
K(PY a>B 4
= qp, a),
Eq. (3.1) gives the joint density of the eigenvalues of the real Wishart matrix; in this case, Eq. (3.3) coincides with Eq. (2.1) of [5]. When /I = 2, d = 1, a=n-pand
qp,
F(p
a, B, 4 = C(P, 4 = l/ii
- i + 1) &
+ p + 1 - i)},
(3.3a)
i=l
Eq. (3.1) gives [8, 91 the joint density of the eigenvalues of the complex Wishart matrix S* with n degrees of freedom and E(S*) = nl, . In the remainder of this section, we will assume that ci < ..* < c, are the eigenvalues of S*. Now, let {a, < ... < ai- be a subset of (1, 2 ,..., p) and let {ti < ... < tDmj+i} be its complementary set. Similarly, let LX,,be a subset (with one element) of (t1 < ... < tP-j+l} and let {/3i < ... < psPj} be its complementary set. Also, let {k, < . . * < kjdl} be a subset of {I,..., p} and (8, < ... < 6,~,+i} be its complementary set. In addition, (Yeis a subset (with one element) of (6, < ... < 6,-j+l}, and (or < ... < yP-?) is its complementary set. Krishnaiah [lo] derived the following expressions for the marginal densities of cj: ~F!~a(w) = C( p, a) ~,~,~,~,(
- l)Pai+q+‘ki+U1
exp( - w) w’+%+~I-~
] A 1 1B 1
(3.4) where x1 denotes the summation over all (&) choices of {a, ,..., a+i}, C2 denotes the summation over all (p - j + 1) p ossible choices of “0 , x3 denotes the summation over all (&) possible choices of (kl ,..., kimI), and x4 denotes the 6831413-2
270
KFtISHNAIAH
AND
SCHUURMANN
summation over all (p - j + 1) possible choices of aI . Also, A = (a,,) and B = (bgh), where w agh = exp( -2) z”+~‘+~~-~dz, I0
g,h = I,..., j-
4,,,= j+* exp( -2)
g,h=l
to
~~+~~+“~-a dz,
1,
,...,p -j.
When j = 1, Eq. (3.5) involves only two summations, namely, Cz and C, and 1A 1 does not appear; a similar simplification occurs when j = p. Alternatively, we make the transformation g, = c,/cj (i = I,..., j - 1, j + I,..., p) and c, = ci in Eq. (3.1), with /3 = 2 and K(p, Y, /I, d) = C(p, I), and integrate ,‘&+I ,"',gp * Using Lemmas 2.2 and 2.3 of [lo] we obtain the out g1 ,-V&-l following expression for the marginal densities of cj . Tt,\(w) = ~(p, Y) exp(-w)
w~‘~+~‘-‘C~C~(-~)~‘~+~~~ I A, I I As I (3.5)
for j = 1,2 ,..., p, where {a, < ... < aj-r} is a subset of (1, 2 ,..., p - l} and {tr < *. . < tDei} is the set complementary to it while {k, < *a. < A,-r} is also a subset of (1, 2,..., p - 1) and (6, < .** < S,-j+,} is complementary to (4 ,..., k,-l}. Then x1 denotes the sum over all (7::) possible choices for {al ,..., aimI} and & denotes the sum over all possible choices for (k, ,..., kjwl}. The elements of the matrices A, and A, are al,gh
=
s03 2
a+tg+8A-2( 1 - z)” edwr dz,
g, h = 1,2 ,..., p - j,
1
1 a2.,h
=
z-~+~~-~( 1 - 2)” e-” dz,
s
g, h = 1,2 ,..., j -
1.
0
When j = 1 or p, 1A, 1or I A, I is set equal to 1, respectively, the sign is positive and neither summation occurs. Starting from Eq. (3.4) or from Eq. (3.5) we observe, after simplification, that the densities of c, (j = l,..., p) are of the following form. hi 7$(w) = C(p, a) exp( -w) ~l”(~)+‘)--lx 1 i-1
f?ij
exp(-%iw) W%l
.
(3.6)
It is tedious to obtain explicit algebraic expressions for the constants dtj , e, and tij , but they can be evaluated with the help of a computer when p is not
LATENT ROOTS OF THE COVARIANCE MATRIX large. Now, using Eqs. (3.3) and (3.6), we obtain expressions for the densities of wi (i = l,..., p): l&w)
= r(p(p
+ a)) C(p, a) f
271
after inversion
the following
eijw”(“+‘)-2
i=l
X
(((1 - (CQ + l)w)~j-l)/w”ij-‘(tij
where (CC)+ is equal to x or 0, accordingly, Using Eq. (3.3), it is seen’ that l&x)
= X”(“++sf;-l,r,2((1
x
-
(3.7)
as x > 0 or x < 0.
- X)/X)
W(P + I)) D-1 [ W) r(y + 1) izl nl qp
qi + 2) r(Y + i) + y + 1 - q qfJ + y + 1 + i) 11
wherefL.,(x >is t he density of T = Cr=, wi and the joint wa (wl > 0, wg < 1) is fi [+
l)!),
density
of wI < ..* <
+ n + 4 +.W(n +A W t-i) WI
x fi {W
i=l
i>j
was given in [lo]
for the distribution
of T in the general case.
4. RATIOS OF THE EXTREME ROOTS OF THE REAL AND COMPLEX WISHART MATRICES Let Then,
fi,
= Ii/l, where the joint the cumulative distribution
density of 1, < *.. < 1, is given by Eq. (2.1). function (c.d.f.) of fi, is given by
where D: c < fi, < ...
< c] = 1 - k(p,
Y)
somexp(-Z1,/2)
joint
1y2)-’
x /4&P - 1,R-4l,...,p - 21, c, 1)4,
(4.2)
272
KRISHNAIAH
AND
SCHUURMANN
where p(#; p - 1, (0, I,..., p - 2}, c, l} is defined by Eqs. (2.4) and (2.5) and (6(x) = xr( 1 - x) Zxp(--xZp/2). When r is an integer, it is seen that p( ...) is of the following form. ~(4; p -
1, (0, l,..., p - 2}, c, 1) = 5 dicri exp{--l,(cs,
+ ti)>/ZEi.
(4.3)
i=l
Whenp is not large, the coefficients di , ri , si , ti and vi can be computed explicitly using a computer. Using Eq. (4.3) in Eq. (4.2), we obtain I’&,
< c] = 1 - k(p, T) F dic+‘T((np/2) - ui),‘(csi + ti + l)((ns’2)-v~! i=l
(4.4)
When r is a half-integer, the c.d.f. of fr, is of the same form as the right side of Eq. (4.4) after IV* is replaced with co. The expression given in this paper for the c.d.f. of fr, is better, from computational point of view, than the corresponding expressions given in [22, 241. Next, let gi, = Q/C, where the joint density of c, < *.. < cp is given by Eq. (3.1) with /? = 2, d = 1, a = n - p and K (p, a, /3, d) given by Eq. (3.3a). Then, the c.d.f. of g,, is given by
%, G c*1= 1 - wd4~m is, ***fh2klP
~**~7gD-l.D~
4
jp&*j
4
(4.5) where R: c* < gr, < .**
ctPA11A 1dc,
(4.6)
where A = (ao) and 1
au =
sC*
exp( -c-)
~n--P+i+j-t(1 - x)” dx.
After simplification, it is seen that 1 A ) is of the following form. 1A 1 = Fdi*c*li*
exp{-c,(c*s,*
+ ti*)}/cii*.
(4.7)
i-1
When p is not large, the constants d6*, TV*, si*, ti* and vi* can be computed using a computer. Now using Eqs. (4.6) and (4.7) we get
qnp - vi*) Pkm G c*l = 1- C(P,Y)F 4*c*li* (1 + ti* + c*s,*yp--vd+ * * i=l
(4.8)
LATENT 5.
ROOTS
OF THE
CONSTRUCTION
COVARIANCE OF THE
MATRIX
273
TABLES
The coefficients dij , qij and mii in Eq. (2.8) and the coefficients eij , cyij and tij in Eq. (3.7) were evaluated with a computer. Then, the values of b and b* were computed for cy = 0.990, 0.975, 0.950, 0.900, 0.750, 0.500, 0.250, 0.100, 0.050, 0.025, 0.010, p = 3(1)6 and different values of T, a and j, where
i
)$(w)
dw = (1 - LX),
s
ob*h2ja(w) dw = (1 - cu),
(5.1) (5.2)
and f:,‘,(w) and h!,:(w) are given by Eqs. (2.8) and (3.7), respectively. The values of b* are computed for p = 3, 4, j = l(l)p whereas the computations were done forj = 1, 2, (p - l), p forp = 5, 6. The computations of b were also done for the above values of p and j except for the cases j = 1, p, since these cases were covered in [20]. As a check for the accuracy of the tables, we computed a few entries in the tables by simulation and these values compare favorably with the corresponding values in the tables. Due to lack of space, we give only some of the values computed. For more extensive tables, the reader is referred to a technical report (ARL 74-0026) by the authors. In Table I, we give the values of b for p = 3(1)6j = 2, p - 1 and different values of Y. The values of b* are given in Table II for p = 3( 1)6, j = 1, 2, p - 1, p and different values of a and 0~. Next, the coefficients di , ri , vi , si , and ti in Eq. (4.3) and di*, ri*, vi*, si*, and ti* in Eq. (4.7) were evaluated using a computer. Then, the values of c and c* were computed for OL= 0.990, 0.975, 0.950, 0.900, 0.750, 0.500, 0.250, 0.100, 0.050, 0.025, 0.010, p = 2(1)6 and different values of n where WI,
< cl = (1 - 4
(5.3)
< c*] = (1 - IX).
(5.4)
and P[gl,
We used the expressions in Eqs. (4.4) and (4.8) to compute the values of c and c*, respectively. As a check for the accuracy of these values, we computed some of the entries by simulation. The values computed by simulation compare favorably with the corresponding values given in this paper. The values of c are given in [24] for a few values of the p, Y and 0~. Some of these values are computed using our program and our computations agree with those given in [24]. This gives an additional check on the accuracy of the values of c given in this paper. Table III gives the values of c for OL= 0.95, 0.99, p = 3(1)5 and a
.3521
.3467
.3346
.3315
4002
.3970
.3940
.3915
.3892
.3872
.3853
.3837
.3822
.3796
-3764
.3723
.3693
2
3
4
5
6
I
8
9
10
12
15
20
25
.2784
.2700
.3182
.3111
.2853
.2910
.2943
.2980
.3023
.3071
.3127
.3193
.3272
.3370
.3494
.3660
j = p -
.3240
.3287
.3381
.3421
.3586
.3666
.3766
1
.3895
.4054
.4035
0
1
LY = 0.05
Percentage
.2661
.2561
.2701
.2746
-2797
.2857
.2926
.3010
.3113
.3243
.3416
1
Points
of the Individual
.3850
.3895
.3955
4002
.4041
.4064
4088
.4116
.4147
.4183
.4224
.4272
.4328
.4395
.4472
.3264
.3354
.3426
.3486
.3521
.3560
.3604
.3654
.3712
.3780
.3861
.3961
.4087
.4251
.2834
.2936
,301s
.3086
.3126
.3171
.3221
.3279
.3346
.3424
.3519
.3635
.3782
.3977
.2783
.2824
.2871
.2923
.2983
.3052
.3134
.3231
.3350
-3501
.3700
(+= 0.01 j = p - 1
of the Ratios
TABLE Roots
I
.2829
.2766
.2674
.2593
.2522
.2479
.2429
.2369
.2298
.2210
.2099
.1952
.1749
.1446
.0946
.1776
.1674
.1589
.1515
.1471
.1421
.1364
.1297
.1217
.1120
.lOOO
.0846
.0640
.0359
0.95
Trace
a =
to the
.1257
.1161
.1084
.I018
.0980
.0936
.0888
.0831
.0766
.0690
.0598
.0486
.0348
.0178
j =
2
Wishart
.0732
.0699
.0662
.0621
.0575
.0523
XI462
.0392
.0309
.0212
.0102
of the Real
.2637
.2553
.2432
.2328
.2236
.2180
.2117
.2042
.1953
.1845
.1711
.1540
.1311
.0991
.0528
0.99
.1627
.1509
.1411
.1328
.I279
.1224
.1160
.1087
.lOOl
.0899
.0775
.0621
.0429
.0196
(Y =
Matrix
.1147
.1043
.0958
.0888
.0802 .0847
.0751
.0693
.0626
.0550
a460
JO96 .0231 .0354
j = 2
.0636
.0565 .0602
.0524
.0477
.0425
.0367
.0300
JO55 .0140 .0224
ii ’
s
s
g
5
m
B ii Eg
2
.5617
.5401
.5204
15
25
.5903
12
20
.6209
6095
9
.6336
8
10
.6649
5482
6
7
.7080
.I364
3
.6846
.I718
2
4
.8174
1
5
.8777
.4263
4449
.4707
.4919
.5097
.5201
.5318
.5451
.5603
.5781
.5989
.6241
.6550
.6942
.7459
p=3~=4~=5~=6
0
r
a = 0.05
.3820
.4057
.4249
4410
.4503
.4608
.4725
.4859
.5013
.5193
.5406
.5664
.5984
.6394
j = p
Percentage
.3903
.3986
.4079
.4182
.4299
.4432
.4586
.4766
.4980
.5240
.5565
Points
.3675
.3699
.3730
.3752
.3768
-3118
.3787
.3197
.3807
.3817
.3827
.3834
.3831
.3829
.3788
a =
.3120
.3172
.3241
.3294
.3337
.3362
.3388
.3418
.3450
.3487
.3528
.3515
.3628
.3689
j
.2891
.2957
.3011
.3042
.3076
.3113
.3155
.3201
.3254
.3315
.3385
.3468
.3567
= p 1
.2862
.2906
.2956
.3012
.3076
.3150
.3237
.3341
of the Individual
.3758
0.05
of the Ratios
.2604
.2517
.2440
.2393
.2340
.2278
a2204
.2115
.2004
.1864
.1678
.1420
.2716
.1734
.1653
.1541
.1449
.1371
.1326
.I275
.1218
.1152
.1076
.0986
.0879
.0749
.0588
.0384
a = 0.95
to the Trace
.1037
II
.2706
Roots
TABLE
.1018
.0937
.0871
.0833
SO792
.0745
.0694
.0636
.0569
.0493
.0304
.0186
j =
of the 2
.0597
.0566
.0532
.0496
.0455
.0411
.0362
.0307
.0246
.0178
.0105
Complex
.17099
.15652
.13725
.10982 .12211
.10283
.08667 .09514
.07727
.05514 .06681
.01354 .02801 .04217
.00213
p=3
Wishart 0.95
.10964
.09838
.08382
.06404 .07276
.05919
.04830 .05395
.04219
.02850 .03559
Xl0599 .01326 .02098
JO085
p=4
OL =
Matrix
.06774
.05636
.04149 .04796
.03795
.03021 .03420
.02597
.01685 .02151
.00318 .00737 .01207
.00043
p=5
j =
1
.02875
.02608
.02034 .02328
.01727
.01085 .01410
.00190 .00453 .00761
.00024
p=6
5 j;
$
fs g
z
ii 8
%
8
B
2
5
276
KRISHNAIAH
AND
SCHUURMANN
TABLE Percentage
a. ___~__
0.99
Points
0.95
of the Ratio of the Smallest Root the Real Wishart Matrix
7 I-
P n
III to the
Largest
Root
of
0.99
a
0.95
a
0.99
0.95
a
l- ___P n
0.99
0.95
P
n
3
24
.I883
.2443
3
36
.2648
.3241
3
48
.3209
.3806
4
27
.1598
.2034
3
26
.2030
.2599
3
38
.2753
.3348
3
50
.3289
.3886
5
14
.0398
.0608
3
28
.2168
.2745
3
40
.2853
.3449
4
19
.I033
.I407
5
16
.0538
.0780
3
30
.2298
.2880
3
42
.2947
.3545
4
21
.1185
.1581
5
18
.0676
.0945
3
32
.2421
.3008
3
44
.3039
.3636
4
23
.I330
.1741
3
34
.2531
.3128
3
46
.3126
.3723
4
25
.1468
.1892
0.99
0.95
.0014
-
TABLE
P ?l
IV
Percentage Points of the Ratio of the Smallest Root Largest Root of the Complex Wishart Matrix
to the
a
0.99
0.95
P n
a
0.99
0.95
P ?z
OL
0.99
0.95
a
~-
P n
2
2
.0017
.008C
2
40
.4632
.5293
P 3
n 14
.I598
.2043
4
4
.0003
2
3
.0192
.0460
2
50
.5039
.5671
3
15
.I721
.2175
4
5
.0041
.0099
2
4
.0465
0.875
2
60
.5359
.5967
3
16
.I838
.2301
4
6
.0123
0.230
2
5
.0751
.I254
2
80
.5838
.6402
3
17
.I950
.2419
4
7
.0227
.0377
2
6
.1025
.I590
3
3
.0006
.0027
3
18
.20>6
.2531
4
8
.0344
.0529
2
7
.1278
.1886
3
4
.0077
.0184
1 3
19
.2159
.2638
4
9
.0466
.0680
20
2
8
.I511
.2150
3
5
.0212
.0399
3
.2256
.2739
4
10
.0588
.0826
2
9
.1725
.2386
3
6
.0375
.0622
3
21
.2350
.2836
4
11
.0709
.0968
2
10
.1922
.2600
3
7
.0545
.0841
3
22
.2439
.2928
4
12
.0827
.1103
2
12
.2273
.2971
3
8
.0715
.1047
3
23
.2525
.3017
4
13
.0942
.1234
9
.0880
.1242
3
24
.2608
.3102
5
5
.OOOl
.0008
14
.2576
.3284
3
2
17
.2962
.3676
3
10
.I037
.I424
3
25
.2688
.3183
5
6
.0026
.0062
2
20
.3287
.3999
3
11
.I188
.1594
3
26
.2765
.3261
5
7
.0079
.0149
2
25
.3729
.4432
3
12
.I332
.I753
3
27
.2839
.3335
5
8
.0153
.0253
2
30
.4086
.4775
3
13
.1468
.1902
3
28
.2910
.3408
5
9
.0238
.0365
2
-
LATENT
few values
of 71. The
ROOTS
values
OF
THE
COVARIANCE
of c* are given
277
MATRIX
in Table
IV for (Y = 0.95, 0.99, to a technical report (unpublished) by the authors for the values of c and c* for other values of 01. All the computations were done using CDC 6600 and the values of c and c* may differ from actual values by at most one unit in the last decimal.
p = 3(1)5 and a few values of n. The reader is referred
6.
APPLICATIONS
Let the columns of X: p x n be distributed as a multivariate normal with zero means and covariance matrix Z. Also, let Zi < ... < I, be the eigenvalues ofS=XX’whereasX, <*.. < X, denote the eigenvalues of 2. In addition, let
H: X, = ... = &, Hij : hi = ,ii ,
P-1 A, = U 4,~
A, = & > hi ,
P-l
3
A2 = u Api
i=l
i=l
and A, = U&j Aij . Also, letfii = Zi/Zj . We will review below some procedures proposed by Krishnaiah and Waikar [15, 161 for testing H against A, , A, and A,; these procedures are useful in drawing inference on subhypotheses of the form Hii . The distribution of f,i is useful in the applications of these test procedures. First, we discuss testing the hypotheses H,, , Hs2 ,..., HP,,-, and H simultaneously against the alternatives A,, , A,, ,..., A,,,-, and A, . In this case, we accept or reject Hi+l,i (i = l,..., p - 1) according as
where P[f<+l,i
<
a,,;
i
The joint distribution of fil ,..., a,, are not yet tabulated. But p[fi+l,i
<
%rr
=
I ye.-)
-
1
H] = (1 - a).
)
was given
f,-l,,
, i =
p
P
I,...,
-
I]
in [l5]
>
p[fp,
but the critical
<
+J.
(6.1) values
(6.2)
So, an upper bound on a,, can be obtained using the tables of the distribution of fQl. Next, consider testing the hypotheses Hgl ,..., HP,+, , and H simultaneously against the alternatives A,, ,..., A,,,-, and A, simultaneously. In this case, we accept or reject HDi according as
for practical purposes, we may choose (say) a2= . Then, u2a is given by P[fm
<
cza
the
I
critical
values
HI = (1 - a).
uIpei to be equal to
(6.3)
278
KRISHNAIAH
AND
SCHUURMANN
The total hypothesis H is accepted or rejected according as f,r 5 Us . If we test the hypotheses Hcj (; > j) and H simultaneously against Aij and A, , we accept or reject Hij according as
where
(6.4) The total hypothesis H is accepted or rejected according asfDl 5 a, . We now propose alternative procedures for testing H when some of the roots are known to be equal. Now, let pX = xi”=, Xi , Hi: hi = A, A, = u*crr [hi < A] and A, = uy=, [hi > A]. Also, let ui = ZJCy=r Zj for i = 1, 2,..., p. We will first consider the problem of testing H against A, when X, = ... = X, . In this case, we accept H if ui 3 b, and reject it otherwise
i = t,...,p -
1
where
P[ui > b,.; i = t,..., p - 1 1 H] = (1 - a).
(6.5)
If H is rejected, then we accept or reject Hi (i = t,...,p - 1) according as ui 2 bi, . If we choose the constants b, to be equal to b, (say), then the critical values b, can be obtained from the tables of the distribution of ut (under H). The hypothesis H can be also tested against A, by using any individual 4l(li + ... + I,), (i = l,..., t), as a test statistic. Next, let At = ... = X, . In this case, we accept H against A, if I(i
and reject it otherwise
d
Cia
for
i = l,..., t,
where
P[ui < ciu; i = l,..., t 1 H] = (1 - 0~).
(6.6)
If H is rejected, the hypothesis Hi (i = l,..., t) is accepted or rejected according as ui 5 ciu . If we choose the critical values to be equal to (say) c, , then c, can be obtained from Table I. The hypothesis H can be also tested against A, by using any individual li/(li + ... + Zr), (i = t,..., p), as a test statistic. hj < i Cj”=, hj], When& = ... = X, , if we wish to test H against u:Ii [p $, we may use (Zr + e.1 + I,)/(/, + 1.. + Z,) as a test statistic. The same test statistic may be also used for testing H against ul=a [p &, Xj > i Cral h,] when A, = ... = X, . One can similarly propose procedures for testing H against other restricted alternatives using Roy’s union intersection principle [ 191.
LATENT
ROOTS
OF THE
COVARIANCE
MATRIX
279
The hypothesis of the equality of the latent roots of the covariance matrix of the complex multivariate normal distribution against different alternatives can be tested by using procedures analogous to those discussed above for the real case. If we wish to test the hypothesis that /\r = ... = X, = A (where X is known) against UL, [A, < h] or lJb, [hi > X], one can use the tests based on the individual roots. Similarly, if one wishes to test H against & [Xi # d], one may use the joint density of the extreme roots. The reader is referred to [2, 3, 12, 13, 15, 211 (and some of the references in those papers) for details of these procedures and tables useful in the applications of these procedures in the real and complex cases. The procedures discussed so far for testing the hypothesis of the equality of the latent roots of the covariance matrix can be used even when the underlying populations are not real or complex multivariate normal. But the distribution problems in this case are more complicated. Procedures based on the differences (instead of the ratios) of the roots for testing the hypothesis of the equality of the roots of covariance matrix will be discussed by one of the authors in a subsequent communication. We will now discuss some applications of the procedures discussed above. samples drawn from a p-variate population Let xi ,..., xN be N independent with the cumulative distribution function (c.d.f) F(x). Also, let Fj(x), (j = 0, l,..., k), be the c.d.f. of the multivariate normal with mean vector pi and covariance matrix ~31, . Let H,, denote the hypothesis that F(x) is of the form F,-,(x) and let A, denote the alternative hypothesis that F(x) is of the form where If=, ri = 1, rrj > 0. The population covariance +-1(x) + ... + ~fl~(x) matrix of xi’s is Z = #I, + Q where 52 = xj”=, ~~j( pj - t.~)(t~,~- t.~)’ is assumed to be of rank (k - 1) and p = x:j”=, xjt+ . The problem of testing Ho is equivalent to testing for the equality of the roots of .Z when the smallest (p - k + 1) roots are equal. The alternative hypothesis A, may be restricted by imposing some conditions on the roots of Q. So, the procedures discussed before for testing H against different alternatives, may be used to test Ho. If H,, is rejected, we can use these procedures to test various subhypotheses on the roots of Z. Alternative methods of testing H,, against A, were considered by Bryant [ 11. Here we note that the sample sums of squares and cross-products (SP) matrix is distributed as the central Wishart matrix with N - I degrees of freedom when Ho is true. If A, is true, the SP matrix is distributed (see [l]) as a linear combination of the non-central Wishart matrices. The problem of testing H,, against A, is useful in the area of signal detection (see [4]). Next, let C = c2[( 1 - p) I + pee’], where e’ = (I,..., 1) and .Z is the covariance matrix of the multivariate normal distribution. If p < 0, then h, = ... = h,_i = u2[1 + (p - l)p] and &, = u2(l - p), where /\r < ...
280
KRISHNAIAH
AND
SCHUURMANN
are the eigenvalues of L’. Here, the hypothesis of p = 0 is equivalent to the hypothesis that h, = ... = h, . So, the procedures discussed before may be used for testing the hypothesis that p = 0. Next, let us assume that p > 0. Then X, = a”(1 - p) and ha = ... = h, = a*[1 + (p - I)p]. In this case, the hypothesis of p = 0 may be tested using the procedures discussed before. Consider the following model considered by Gollob [6] and Mandel [17]. yij = P + R, + Cj + ; hwUji
+ ~3 ,
i = 1).. . , s, j = 1,. . ., t,
Z=l
where yii denotes the observed value in ith row andjth column, K < min(s, t) - 1, ‘& Ri = & Ci = cb, uil = xi=, zljl = 0 and h, > ... > /\1 >, 0. Also, xi=, Z& = & Z$ = 1 and XI=, zliluil’ = Ci=, vilvjl’ = 0 for E # I’. In addition, cij are distributed independently as normal variates with zero means and variances one. Now, let W = K,Y (I” - f Js) Y’Kt’, where Y = ( yij), Js is a matrix of order s x s whose elements are one and K, is a matrix of order (t - 1) x t such that KtKt’ = I,-, and K,‘K, = It - (l/t) Jt . Without loss of generality, let us assume that t < s. Then, it is known (e.g., see [26]) that W is a Wishart matrix with (s - 1) degrees of freedom and E(W/(s - 1)) = Z* = u*lt-l + 52 where Sz = (I/(s - 1)) K,lT’K,’ , r = (vij) and vij = ‘&=r X,uilvjl . The problem of testing the hypothesis that h, = ~1.= X, = 0 is equivalent to the problem of testing the hypothesis that the eigenvalues of Q are equal since Q is not of full rank. So, the methods discussed in this section can be used to test the hypothesis that X, = *.. = h, = 0 against various alternatives; when this hypothesis is rejected, we can also draw inference as to which of the hi’s are significantly different from zero. Here, we note that it is known that the K largest eigenvalues of W are the least square estimates of h, ,..., h, . The problem of testing the hypotheses Ai = 0 was considered by Gollob [6], Mandell [17] and Yochmowitz [25] from other points of view. Next, let X’(t) = (X1(t),..., XJt)) be a p-dimensional stationary, Gaussian time series with zero means and covariance matrix R(s) = (Rij(s)) and spectral density matrix F(w) = (fii(w)) where
&j(s) = W&(t) Xi(t + 41, fij(w) = &
2 exp(--isw) Rij(s). *=--m
LATENT
ROOTS
OF THE
COVARIANCE
MATRIX
281
Let p1 < ... < pD (pl > 0, ~1~< 00) be the eigenvalues of F(w) and let us assume that
The observed time series consists of a single record of length T. A well-known estimate of F(w) is p(w) = (fij(w)), where Z* denotes the complex conjugate of 2, fidw) =
(2n :
f Iii (w + -g, 1) m=--n
and n is a suitably chosen integer. (For a review of the literature on multiple time series, the reader is referred to Parzen [18].) It is known (see [7, 231) that (2n + 1) E(w) is approximately distributed as complex Wishart matrix with (2n+ 1) degrees of freedom and E(F(w)) = F(w). So, we can test the hypothesis that F(w) = ~1~ against different alternatives by using procedures analogous to those discussed in this section for testing the hypothesis that the eigenvalues by the covariance matrix of the multivariate normal are equal.
REFERENCES [I]
P. G. (1971). Tests and estimates of the number of components in a mixture. Rept, No. 27, Department of Statistics, Stanford University. 123 CLEMM, D. S., CHATTOPADHYAY, A. K., AND KRISHNAIAH, P. R. (1973). Upper percentage points of the individual roots of the Wishart matrix. Sankhya Ser. B 35 325-338. [3] CLEMM, D. S., KRISHNAIAH, P. R., AND WAIKAR, V. B. (1971). Tables for the extreme roots of the Wishart matrix. J. Statist. Co?@. Simul. 2 65-92. [4] COOPER, D. B. AND COOPER, P. W. (1964). Nonsupervised adaptive signal detection and pattern recognition. Information and Control 7 416-444. [5] DAVIS, A. W. (1972). On the ratios of the individual latent roots to the trace of a Wishart matrix. J. Multivariate Anal. 2 4404l3. 61 GOLLOB, H. F. (1968). A statistical model which combines features of factor analytic and analysis of variance techniques. Psychometrika 33 73-116. [7] GOODMAN, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution. Ann. Math. Statist. 34 152-177. [8] JAMES, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 475-501. BRYANT,
Tech.
282 [9]
KRISHNAIAH
[ll]
[12]
[13]
[14] [I51
[16]
[17]
[18] [19] [20]
[21] [22] [23] [24]
[25] [26]
SCHUURMANN
C. G. (1965). Classical statistical analysis based on a certain multivariate Gaussian distribution. Ann. Math. Statist. 36 98-114. KRISHNAIAH, P. R. (1973). On the exact distributions of the test statistics based on the eigenvalues of complex random matrices. ARL 73-0178, Aerospace Research Laboratories, Wright-Patterson AFB, Ohio; (also in Proceedings of the 39th Session of the International Statistical Institute held at Vienna). KRISHNAIAH, P. R. AND CHANG, T. C. (1972). On the exact distributions of the extreme roots of the Wishart and MANOVA matrices. 1. Multivariate Anal. 1 108-117. KRISHNAIAH, P. R. AND SCHUURMANN, F. J. (1974). On the exact distributions of the individual roots of the complex Wishart and multivariate beta matrices. ARL 74-0063. Aerospace Research Laboratories, Wright-Patterson AFB, Ohio. KRISNAIAH, P. R., AND SCHUURMAN, F. J. (1974). On the exact joint distributions of the extreme roots of the complex Wishart and multivariate beta matrices. ARL 74-0111. Aerospace Research Laboratories, Wright-Patterson AFB, Ohio. KRISHNAIAH, P. R. AND WAIKAR, V. B. (1971). Exact distributions of any few ordered roots of a class of random matrices. /. Multivariate Anal. 1 308-315. KRISHNAIAH, P. R. AND WAIKAR, V. B. (1971). S’ imultaneous tests for equality of latent roots against certain alternatives-I. Ann. Inst. Statist. Math. 23 451-468; (also ARL Tech. Report 69-0119, 1969). KRISHNAIAH, P. R. AND WAIKAR, V. B. (1972). Simultaneous tests for equality of latent roots against certain alternatives-II. Ann. Inst. Statist. Math. 24 81-85; (also ARL Tech. Report 69-0178, 1969). MANDEL, J. (1969). The partitioning of interaction in analysis of variance. J. Res. Nut. Bur. Stand. 73B 309-328. PARZEN, E. (1969). Multiple time series modeling. In Multiwuriute Analysis--II (P. R. Krishnaiah, Ed.). Academic Press. ROY, S. N. (1957). Some aspects of multiwuriute analysis. Wiley, New York. SCHUURMANN, F. J., KRISHNAIAH, P. R., AND CHATTOPADHYAY, A. K. (1973). On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix. J. Multivariate Anal. 3 445-453. SCHUURMANN, F. J. AND WAIKAR, V. B. (1974). Upper percentage points of the individual roots of the complex Wishart matrix. (to appear) Sunkhyu Ser. B 36. SUGIYAMA, T. (1970). Joint distribution of the extreme roots of a covariance matrix. Ann. Math. Statist. 41 655-657. WAHBA, G. (1968). On the distribution of some statistics useful in the analysis of jointly stationary time series. Ann. Math. Statist. 39 1849-1862. WAIKAR, V. B. AND SCHUURMANN, F. J. (1973). Exact joint density of the largest and smallest roots of the Wishart and MANOVA matrices. Utilitus Muthemuticu 4 253-260. YOCHMOWITZ, M. (1974). A note on partitioning the interaction in a two-way model with no replication. ZMS Bull. 3 80. JOHNSON, D. E. AND GRAYBILL, F. A. (1972). An analysis of a two-way model with interaction and no replication. J. Amer. Statist. Assoc. 67 862-868. KHATRI,
complex
[lo]
AND
OF MULTIVARIATE
4,265-282
ANALYSIS
(1974)
On the Evaluation of Some Distributions that Arise Simultaneous Tests for the Equality of the Latent Roots of the Covariance Matrix P. R. Aerospace
Research
KRISHNAIAH Laboratories,
AND
F. J.
Wright-Patterson
in
SCHUURMANN* Air
Force
Base,
Ohio
45433
In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given.
1.
INTRODUCTION
There are several physical situations where the experimenter is interested in testing for the equality of the latent roots of the covariance matrices of real and complex multivarite normal populations. Krishnaiah and Waikar [15, 161 proposed certain procedures for testing the hypothesis of the equality of the latent roots of the covariance matrix of the real multivariate normal population. Similar procedures can be used in the complex case also. These procedures are also useful in drawing inference on certain subhypotheses when the total hypothesis is rejected. The distributions of the ratios of the extreme roots and the ratios of the individual roots to the sum of the roots of the real and complex Wishart matrices arise in the procedures proposed in [15, 161 and some other test procedures discussed in this paper. The purpose of this paper is to investigate the evaluation of the probability integrals of the above distributions and give percentage points which are useful in the application of the above procedures. AMS 1970 subject classifications: Primary 62HlO; Secondary 62E15. Key words and phrases: Exact distributions, ratios of the roots, Wishart matrix, multiple time series, tests for additivity, simultaneous tests, mixture of populations. * The work of this author was done at the Aerospace Research Laboratories while in the capacity of a Technology Incorporated Visiting Research Associate under Contract F33615-71-C-1463. This work may be reproduced in part or whole for any purpose of the United States Government. The present address of this author is Miami University, Oxford, Ohio.
265 Copyright All rights
0 1974 by Academic Press, Inc. of reproduction in any form reserved.
266
KRISHNAIAH
AND
SCHUURMANN
In the case of the real Wishart matrix, Davis [5] established a useful relationship between the Laplace transformations of the ratios of the individual roots to the sum of the roots and the densities of the above individual roots. Using this relationship, and the expressions in [1 1] for the densities of the extreme roots of the real Wishart matrix, Schuurmann, Krishnaiah, and Chattopadhyay [20] derived expressions for the distributions of the ratios of the extreme roots to the trace of the real Wishart matrix; they have also constructed tables for these distributions. In Section 2 of this paper, the authors derive expressions for the distributions of the ratios of the intermediate roots to the trace of the real Wishart matrix by using Eq. (2.1) of [5] and the expressions given in Krishna& and Waikar [14] for the intermediate roots of the real Wishart matrix. Tables for these distributions are also constructed for some values of the parameters. In Section 3, the authors obtain a relationship between the Laplace transformations of the ratios of the individual roots to the trace of the complex Wishart matrix and the distributions of the individual roots of this matrix. Using this relationship and the expressions given in Krishnaiah [IO] for the densities of the individual roots of the complex Wishart matrix, the authors obtain expressions for the distributions of the ratios of the individual roots to the trace of the above matrix. They have also constructed tables for these distributions. The evaluation of the probability integrals connected with the ratios of the extreme roots of the real and complex Wishart matrices is discussed in Section 4. Tables for the percentage points of these distributions are given in Section 5. Finally, the applications of the distributions considered in this paper are discussed in Section 6.
2. DISTRIBUTIONS
OF THE RATIOS OF THE INTERMEDIATE ROOTS TO THE TRACE OF THE REAL WISHART MATRIX
Let Zi < **. < 1, be the latent roots of the real Wishart matrix S with n degrees of freedom and E(S) = nI, , where I, is the identity matrix of order p x p. Then, the joint density of the above roots is well known [I93 to be
where
0 < Zr < ... < Z, < co, r = (n - p -
k(p,y)=
,+P(1/2)9(2*+~+1)/2 p VW
1)/2 and
+ P + 2 - W) WJ + 1 - i)/2)].
LATENT
ROOTS
OF THE
COVARIANCE
267
MATRIX
In this section, we derive the densities of ua ,..., u,-r where ui = l&Z1 lj for i = l,...,p. Let f,‘fi and gr’,, denote the densities of uj and 1, , respectively. Davis [S] showed that ,Ee{(l + w)(Zrp+p”+p-4)/2~~~(1/(1 = 2T(p(2r
+ w))}
+ p + 1)/2) e’s’2-p(zz+P+1))/2gai~(2F),
(2.2)
where Y(h(w)) denotes the Laplace transformation of h(w). Krishnaiah Waikar [14] showed that the density of li (2 < j < p - 1) is given by gg.(w)
and
= k( p, t) Cl x2 (- 1 )v+~~+~::~:%Jl(W)W”’ x P(&j-
l,{k,
,-e-s k,-d,o,
wu)&;
P-
j,&
,..., Bp-j), W, 4,
(2.3)
where #(w) = exp(-w/2) wr and v = )( jT + j + 2). Also, (K, < ... < R,-l} is a subset of the set of integers (0, l,...,p - l}, {tr < ... < t,-i+l} is the over subset complementary to {k, < ... < k,-,), and zr denotes the summation all (if,) possible choices of {k, < ... < kjpl}. Similarly, CY~is a subset (with one element) of the set of integers {tl ,..., t,-j+l}, {& < ... < j3p--i} is the subset complementary to q , and x2 denotes the summation over all (p - j + 1) possible choices for .q . Also, the p’s are as defined below ~(4; 2~ (4 ,..., k,,), L, U) = 44;
24 (4 >..., k,th L, U)
(2.4)
and ,4k
2 + 1, (4 ,..., k2t+J,L,
U)
2t+1 =
zl
(-l)“+‘F,,(L,
U)
GiO,k
2 + 1, (4 ,..., kzt+J,L,
u)
(2.5)
where 44;
24 (k, 9.0., k,tLL, W
= I(f:@, Uhi=1.....2t11’2,
G,(lcI;2t + 1,{k, ,..., ht+JsL u) = I(fS$L, u)h=l..... cI-l.q+l,..., St+111’2, F,(L, U) = pwds and
Making
the transformations
gi = ZJZj , (i = l,..., j -
I,j
+ I,..., p) and Zj = Zj
KRISHNAIAH AND SCHUURMANN
268
in Eq. (2.1) and integrating out g, ,..., gj-r , gj+r ,..., g, by applying Lemma 2.1 of [14], we obtain the following alternative expression for the density of 4. g:,:(w) = k(p,
Y)
exp(-wj2)
xC(-1)
~(p(~~+J+l)-~)/~
(j(i+2)/2)+~:=:acf(~l;j - 1, {al ,..., aj-l}, 0, 1)
x ~(4~; P -j,
& ,..., hj>, 1, ~0)
(2.6)
where {a, < ... < aj-r} is a subset of (0, l,...,p - 2) and {S, < ... < SPPj} is the subset complementary to {a, ,,.., a&, and C denotes the summation over all ( ;I:) possible choices of (a, < ..* < aiPl}. Also &(x) = exp(-wx/2)(
I - X) ~(P(~~+p+l)-~)/~
and #2(~) = exp(-wx/2)(x - 1) ~(P(~~+p+l)-~)/~.Starting from Eq. (2.3) (or Eq. (2.6)), we observe that g:.\(w) is of the following form gi!T(w) = k( p,
Y)
exp( -w/2) w(‘(~~+‘+~)-~)‘~gl
dij exp( -Tijw)/wmij,
(2.7)
when Y is an integer. Now using Eqs. (2.2) and (2.7), we obtain after inversion the following: f$(z)
= r((p(2Y
+ p + 1) -
2)/2) k(p, Y)2(P(2T+P+1)‘*) (2.8)
where (x)+ is equal to x or 0, accordingly, as x > 0 or x < 0. It is tedious to obtain algebraic expressions for the constants nij , dii and mij . But for fixed values of p and Y (p not very large) they can be computed using a computer. When Y is not an integer,fF,\(w) is of the same form as Eq. (2.8) with N replaced by co. 3. DISTRIBUTIONS OF THE RATIOS OF THE INDIVIDUAL
ROOTS TO THE TRACE OF THE COMPLEX WISHART MATRIX
Consider a class of random matrices whose eigenvalues cr ,..., cg have the joint distribution
fib ,***14 = K(P, a, 8,d) fi [cP exp(-dc,)] n (ci - ci)s i=l
i>j
(3.1)
LATENT
ROOTS
OF
THE
COVARIANCE
269
MATRIX
where0 < ci < ... < c, < co and K(p, a, /3, d) is a normalizing constant. Now, let oi = ci/CLl cj for i = 1, 2,..., p. Also, let h& and yF,L denote the densities of vui and cj , respectively. The joint density of va ,..., v, is seen to be
f&2
9 573 ,..*>
vP)
=
where CL, vi = 1. Starting Davis [5], we obtain
K(
P> a9
from
B9 d)(
l/d)"r(S)
fil
via
Eq. (3.2) and following
n
i>j
(Vi
-
Vj)”
(3.2)
the same lines as in
(3.3)
K(PY a>B 4
= qp, a),
Eq. (3.1) gives the joint density of the eigenvalues of the real Wishart matrix; in this case, Eq. (3.3) coincides with Eq. (2.1) of [5]. When /I = 2, d = 1, a=n-pand
qp,
F(p
a, B, 4 = C(P, 4 = l/ii
- i + 1) &
+ p + 1 - i)},
(3.3a)
i=l
Eq. (3.1) gives [8, 91 the joint density of the eigenvalues of the complex Wishart matrix S* with n degrees of freedom and E(S*) = nl, . In the remainder of this section, we will assume that ci < ..* < c, are the eigenvalues of S*. Now, let {a, < ... < ai- be a subset of (1, 2 ,..., p) and let {ti < ... < tDmj+i} be its complementary set. Similarly, let LX,,be a subset (with one element) of (t1 < ... < tP-j+l} and let {/3i < ... < psPj} be its complementary set. Also, let {k, < . . * < kjdl} be a subset of {I,..., p} and (8, < ... < 6,~,+i} be its complementary set. In addition, (Yeis a subset (with one element) of (6, < ... < 6,-j+l}, and (or < ... < yP-?) is its complementary set. Krishnaiah [lo] derived the following expressions for the marginal densities of cj: ~F!~a(w) = C( p, a) ~,~,~,~,(
- l)Pai+q+‘ki+U1
exp( - w) w’+%+~I-~
] A 1 1B 1
(3.4) where x1 denotes the summation over all (&) choices of {a, ,..., a+i}, C2 denotes the summation over all (p - j + 1) p ossible choices of “0 , x3 denotes the summation over all (&) possible choices of (kl ,..., kimI), and x4 denotes the 6831413-2
270
KFtISHNAIAH
AND
SCHUURMANN
summation over all (p - j + 1) possible choices of aI . Also, A = (a,,) and B = (bgh), where w agh = exp( -2) z”+~‘+~~-~dz, I0
g,h = I,..., j-
4,,,= j+* exp( -2)
g,h=l
to
~~+~~+“~-a dz,
1,
,...,p -j.
When j = 1, Eq. (3.5) involves only two summations, namely, Cz and C, and 1A 1 does not appear; a similar simplification occurs when j = p. Alternatively, we make the transformation g, = c,/cj (i = I,..., j - 1, j + I,..., p) and c, = ci in Eq. (3.1), with /3 = 2 and K(p, Y, /I, d) = C(p, I), and integrate ,‘&+I ,"',gp * Using Lemmas 2.2 and 2.3 of [lo] we obtain the out g1 ,-V&-l following expression for the marginal densities of cj . Tt,\(w) = ~(p, Y) exp(-w)
w~‘~+~‘-‘C~C~(-~)~‘~+~~~ I A, I I As I (3.5)
for j = 1,2 ,..., p, where {a, < ... < aj-r} is a subset of (1, 2 ,..., p - l} and {tr < *. . < tDei} is the set complementary to it while {k, < *a. < A,-r} is also a subset of (1, 2,..., p - 1) and (6, < .** < S,-j+,} is complementary to (4 ,..., k,-l}. Then x1 denotes the sum over all (7::) possible choices for {al ,..., aimI} and & denotes the sum over all possible choices for (k, ,..., kjwl}. The elements of the matrices A, and A, are al,gh
=
s03 2
a+tg+8A-2( 1 - z)” edwr dz,
g, h = 1,2 ,..., p - j,
1
1 a2.,h
=
z-~+~~-~( 1 - 2)” e-” dz,
s
g, h = 1,2 ,..., j -
1.
0
When j = 1 or p, 1A, 1or I A, I is set equal to 1, respectively, the sign is positive and neither summation occurs. Starting from Eq. (3.4) or from Eq. (3.5) we observe, after simplification, that the densities of c, (j = l,..., p) are of the following form. hi 7$(w) = C(p, a) exp( -w) ~l”(~)+‘)--lx 1 i-1
f?ij
exp(-%iw) W%l
.
(3.6)
It is tedious to obtain explicit algebraic expressions for the constants dtj , e, and tij , but they can be evaluated with the help of a computer when p is not
LATENT ROOTS OF THE COVARIANCE MATRIX large. Now, using Eqs. (3.3) and (3.6), we obtain expressions for the densities of wi (i = l,..., p): l&w)
= r(p(p
+ a)) C(p, a) f
271
after inversion
the following
eijw”(“+‘)-2
i=l
X
(((1 - (CQ + l)w)~j-l)/w”ij-‘(tij
where (CC)+ is equal to x or 0, accordingly, Using Eq. (3.3), it is seen’ that l&x)
= X”(“++sf;-l,r,2((1
x
-
(3.7)
as x > 0 or x < 0.
- X)/X)
W(P + I)) D-1 [ W) r(y + 1) izl nl qp
qi + 2) r(Y + i) + y + 1 - q qfJ + y + 1 + i) 11
wherefL.,(x >is t he density of T = Cr=, wi and the joint wa (wl > 0, wg < 1) is fi [+
l)!),
density
of wI < ..* <
+ n + 4 +.W(n +A W t-i) WI
x fi {W
i=l
i>j
was given in [lo]
for the distribution
of T in the general case.
4. RATIOS OF THE EXTREME ROOTS OF THE REAL AND COMPLEX WISHART MATRICES Let Then,
fi,
= Ii/l, where the joint the cumulative distribution
density of 1, < *.. < 1, is given by Eq. (2.1). function (c.d.f.) of fi, is given by
where D: c < fi, < ...
< c] = 1 - k(p,
Y)
somexp(-Z1,/2)
joint
1y2)-’
x /4&P - 1,R-4l,...,p - 21, c, 1)4,
(4.2)
272
KRISHNAIAH
AND
SCHUURMANN
where p(#; p - 1, (0, I,..., p - 2}, c, l} is defined by Eqs. (2.4) and (2.5) and (6(x) = xr( 1 - x) Zxp(--xZp/2). When r is an integer, it is seen that p( ...) is of the following form. ~(4; p -
1, (0, l,..., p - 2}, c, 1) = 5 dicri exp{--l,(cs,
+ ti)>/ZEi.
(4.3)
i=l
Whenp is not large, the coefficients di , ri , si , ti and vi can be computed explicitly using a computer. Using Eq. (4.3) in Eq. (4.2), we obtain I’&,
< c] = 1 - k(p, T) F dic+‘T((np/2) - ui),‘(csi + ti + l)((ns’2)-v~! i=l
(4.4)
When r is a half-integer, the c.d.f. of fr, is of the same form as the right side of Eq. (4.4) after IV* is replaced with co. The expression given in this paper for the c.d.f. of fr, is better, from computational point of view, than the corresponding expressions given in [22, 241. Next, let gi, = Q/C, where the joint density of c, < *.. < cp is given by Eq. (3.1) with /? = 2, d = 1, a = n - p and K (p, a, /3, d) given by Eq. (3.3a). Then, the c.d.f. of g,, is given by
%, G c*1= 1 - wd4~m is, ***fh2klP
~**~7gD-l.D~
4
jp&*j
4
(4.5) where R: c* < gr, < .**
ctPA11A 1dc,
(4.6)
where A = (ao) and 1
au =
sC*
exp( -c-)
~n--P+i+j-t(1 - x)” dx.
After simplification, it is seen that 1 A ) is of the following form. 1A 1 = Fdi*c*li*
exp{-c,(c*s,*
+ ti*)}/cii*.
(4.7)
i-1
When p is not large, the constants d6*, TV*, si*, ti* and vi* can be computed using a computer. Now using Eqs. (4.6) and (4.7) we get
qnp - vi*) Pkm G c*l = 1- C(P,Y)F 4*c*li* (1 + ti* + c*s,*yp--vd+ * * i=l
(4.8)
LATENT 5.
ROOTS
OF THE
CONSTRUCTION
COVARIANCE OF THE
MATRIX
273
TABLES
The coefficients dij , qij and mii in Eq. (2.8) and the coefficients eij , cyij and tij in Eq. (3.7) were evaluated with a computer. Then, the values of b and b* were computed for cy = 0.990, 0.975, 0.950, 0.900, 0.750, 0.500, 0.250, 0.100, 0.050, 0.025, 0.010, p = 3(1)6 and different values of T, a and j, where
i
)$(w)
dw = (1 - LX),
s
ob*h2ja(w) dw = (1 - cu),
(5.1) (5.2)
and f:,‘,(w) and h!,:(w) are given by Eqs. (2.8) and (3.7), respectively. The values of b* are computed for p = 3, 4, j = l(l)p whereas the computations were done forj = 1, 2, (p - l), p forp = 5, 6. The computations of b were also done for the above values of p and j except for the cases j = 1, p, since these cases were covered in [20]. As a check for the accuracy of the tables, we computed a few entries in the tables by simulation and these values compare favorably with the corresponding values in the tables. Due to lack of space, we give only some of the values computed. For more extensive tables, the reader is referred to a technical report (ARL 74-0026) by the authors. In Table I, we give the values of b for p = 3(1)6j = 2, p - 1 and different values of Y. The values of b* are given in Table II for p = 3( 1)6, j = 1, 2, p - 1, p and different values of a and 0~. Next, the coefficients di , ri , vi , si , and ti in Eq. (4.3) and di*, ri*, vi*, si*, and ti* in Eq. (4.7) were evaluated using a computer. Then, the values of c and c* were computed for OL= 0.990, 0.975, 0.950, 0.900, 0.750, 0.500, 0.250, 0.100, 0.050, 0.025, 0.010, p = 2(1)6 and different values of n where WI,
< cl = (1 - 4
(5.3)
< c*] = (1 - IX).
(5.4)
and P[gl,
We used the expressions in Eqs. (4.4) and (4.8) to compute the values of c and c*, respectively. As a check for the accuracy of these values, we computed some of the entries by simulation. The values computed by simulation compare favorably with the corresponding values given in this paper. The values of c are given in [24] for a few values of the p, Y and 0~. Some of these values are computed using our program and our computations agree with those given in [24]. This gives an additional check on the accuracy of the values of c given in this paper. Table III gives the values of c for OL= 0.95, 0.99, p = 3(1)5 and a
.3521
.3467
.3346
.3315
4002
.3970
.3940
.3915
.3892
.3872
.3853
.3837
.3822
.3796
-3764
.3723
.3693
2
3
4
5
6
I
8
9
10
12
15
20
25
.2784
.2700
.3182
.3111
.2853
.2910
.2943
.2980
.3023
.3071
.3127
.3193
.3272
.3370
.3494
.3660
j = p -
.3240
.3287
.3381
.3421
.3586
.3666
.3766
1
.3895
.4054
.4035
0
1
LY = 0.05
Percentage
.2661
.2561
.2701
.2746
-2797
.2857
.2926
.3010
.3113
.3243
.3416
1
Points
of the Individual
.3850
.3895
.3955
4002
.4041
.4064
4088
.4116
.4147
.4183
.4224
.4272
.4328
.4395
.4472
.3264
.3354
.3426
.3486
.3521
.3560
.3604
.3654
.3712
.3780
.3861
.3961
.4087
.4251
.2834
.2936
,301s
.3086
.3126
.3171
.3221
.3279
.3346
.3424
.3519
.3635
.3782
.3977
.2783
.2824
.2871
.2923
.2983
.3052
.3134
.3231
.3350
-3501
.3700
(+= 0.01 j = p - 1
of the Ratios
TABLE Roots
I
.2829
.2766
.2674
.2593
.2522
.2479
.2429
.2369
.2298
.2210
.2099
.1952
.1749
.1446
.0946
.1776
.1674
.1589
.1515
.1471
.1421
.1364
.1297
.1217
.1120
.lOOO
.0846
.0640
.0359
0.95
Trace
a =
to the
.1257
.1161
.1084
.I018
.0980
.0936
.0888
.0831
.0766
.0690
.0598
.0486
.0348
.0178
j =
2
Wishart
.0732
.0699
.0662
.0621
.0575
.0523
XI462
.0392
.0309
.0212
.0102
of the Real
.2637
.2553
.2432
.2328
.2236
.2180
.2117
.2042
.1953
.1845
.1711
.1540
.1311
.0991
.0528
0.99
.1627
.1509
.1411
.1328
.I279
.1224
.1160
.1087
.lOOl
.0899
.0775
.0621
.0429
.0196
(Y =
Matrix
.1147
.1043
.0958
.0888
.0802 .0847
.0751
.0693
.0626
.0550
a460
JO96 .0231 .0354
j = 2
.0636
.0565 .0602
.0524
.0477
.0425
.0367
.0300
JO55 .0140 .0224
ii ’
s
s
g
5
m
B ii Eg
2
.5617
.5401
.5204
15
25
.5903
12
20
.6209
6095
9
.6336
8
10
.6649
5482
6
7
.7080
.I364
3
.6846
.I718
2
4
.8174
1
5
.8777
.4263
4449
.4707
.4919
.5097
.5201
.5318
.5451
.5603
.5781
.5989
.6241
.6550
.6942
.7459
p=3~=4~=5~=6
0
r
a = 0.05
.3820
.4057
.4249
4410
.4503
.4608
.4725
.4859
.5013
.5193
.5406
.5664
.5984
.6394
j = p
Percentage
.3903
.3986
.4079
.4182
.4299
.4432
.4586
.4766
.4980
.5240
.5565
Points
.3675
.3699
.3730
.3752
.3768
-3118
.3787
.3197
.3807
.3817
.3827
.3834
.3831
.3829
.3788
a =
.3120
.3172
.3241
.3294
.3337
.3362
.3388
.3418
.3450
.3487
.3528
.3515
.3628
.3689
j
.2891
.2957
.3011
.3042
.3076
.3113
.3155
.3201
.3254
.3315
.3385
.3468
.3567
= p 1
.2862
.2906
.2956
.3012
.3076
.3150
.3237
.3341
of the Individual
.3758
0.05
of the Ratios
.2604
.2517
.2440
.2393
.2340
.2278
a2204
.2115
.2004
.1864
.1678
.1420
.2716
.1734
.1653
.1541
.1449
.1371
.1326
.I275
.1218
.1152
.1076
.0986
.0879
.0749
.0588
.0384
a = 0.95
to the Trace
.1037
II
.2706
Roots
TABLE
.1018
.0937
.0871
.0833
SO792
.0745
.0694
.0636
.0569
.0493
.0304
.0186
j =
of the 2
.0597
.0566
.0532
.0496
.0455
.0411
.0362
.0307
.0246
.0178
.0105
Complex
.17099
.15652
.13725
.10982 .12211
.10283
.08667 .09514
.07727
.05514 .06681
.01354 .02801 .04217
.00213
p=3
Wishart 0.95
.10964
.09838
.08382
.06404 .07276
.05919
.04830 .05395
.04219
.02850 .03559
Xl0599 .01326 .02098
JO085
p=4
OL =
Matrix
.06774
.05636
.04149 .04796
.03795
.03021 .03420
.02597
.01685 .02151
.00318 .00737 .01207
.00043
p=5
j =
1
.02875
.02608
.02034 .02328
.01727
.01085 .01410
.00190 .00453 .00761
.00024
p=6
5 j;
$
fs g
z
ii 8
%
8
B
2
5
276
KRISHNAIAH
AND
SCHUURMANN
TABLE Percentage
a. ___~__
0.99
Points
0.95
of the Ratio of the Smallest Root the Real Wishart Matrix
7 I-
P n
III to the
Largest
Root
of
0.99
a
0.95
a
0.99
0.95
a
l- ___P n
0.99
0.95
P
n
3
24
.I883
.2443
3
36
.2648
.3241
3
48
.3209
.3806
4
27
.1598
.2034
3
26
.2030
.2599
3
38
.2753
.3348
3
50
.3289
.3886
5
14
.0398
.0608
3
28
.2168
.2745
3
40
.2853
.3449
4
19
.I033
.I407
5
16
.0538
.0780
3
30
.2298
.2880
3
42
.2947
.3545
4
21
.1185
.1581
5
18
.0676
.0945
3
32
.2421
.3008
3
44
.3039
.3636
4
23
.I330
.1741
3
34
.2531
.3128
3
46
.3126
.3723
4
25
.1468
.1892
0.99
0.95
.0014
-
TABLE
P ?l
IV
Percentage Points of the Ratio of the Smallest Root Largest Root of the Complex Wishart Matrix
to the
a
0.99
0.95
P n
a
0.99
0.95
P ?z
OL
0.99
0.95
a
~-
P n
2
2
.0017
.008C
2
40
.4632
.5293
P 3
n 14
.I598
.2043
4
4
.0003
2
3
.0192
.0460
2
50
.5039
.5671
3
15
.I721
.2175
4
5
.0041
.0099
2
4
.0465
0.875
2
60
.5359
.5967
3
16
.I838
.2301
4
6
.0123
0.230
2
5
.0751
.I254
2
80
.5838
.6402
3
17
.I950
.2419
4
7
.0227
.0377
2
6
.1025
.I590
3
3
.0006
.0027
3
18
.20>6
.2531
4
8
.0344
.0529
2
7
.1278
.1886
3
4
.0077
.0184
1 3
19
.2159
.2638
4
9
.0466
.0680
20
2
8
.I511
.2150
3
5
.0212
.0399
3
.2256
.2739
4
10
.0588
.0826
2
9
.1725
.2386
3
6
.0375
.0622
3
21
.2350
.2836
4
11
.0709
.0968
2
10
.1922
.2600
3
7
.0545
.0841
3
22
.2439
.2928
4
12
.0827
.1103
2
12
.2273
.2971
3
8
.0715
.1047
3
23
.2525
.3017
4
13
.0942
.1234
9
.0880
.1242
3
24
.2608
.3102
5
5
.OOOl
.0008
14
.2576
.3284
3
2
17
.2962
.3676
3
10
.I037
.I424
3
25
.2688
.3183
5
6
.0026
.0062
2
20
.3287
.3999
3
11
.I188
.1594
3
26
.2765
.3261
5
7
.0079
.0149
2
25
.3729
.4432
3
12
.I332
.I753
3
27
.2839
.3335
5
8
.0153
.0253
2
30
.4086
.4775
3
13
.1468
.1902
3
28
.2910
.3408
5
9
.0238
.0365
2
-
LATENT
few values
of 71. The
ROOTS
values
OF
THE
COVARIANCE
of c* are given
277
MATRIX
in Table
IV for (Y = 0.95, 0.99, to a technical report (unpublished) by the authors for the values of c and c* for other values of 01. All the computations were done using CDC 6600 and the values of c and c* may differ from actual values by at most one unit in the last decimal.
p = 3(1)5 and a few values of n. The reader is referred
6.
APPLICATIONS
Let the columns of X: p x n be distributed as a multivariate normal with zero means and covariance matrix Z. Also, let Zi < ... < I, be the eigenvalues ofS=XX’whereasX, <*.. < X, denote the eigenvalues of 2. In addition, let
H: X, = ... = &, Hij : hi = ,ii ,
P-1 A, = U 4,~
A, = & > hi ,
P-l
3
A2 = u Api
i=l
i=l
and A, = U&j Aij . Also, letfii = Zi/Zj . We will review below some procedures proposed by Krishnaiah and Waikar [15, 161 for testing H against A, , A, and A,; these procedures are useful in drawing inference on subhypotheses of the form Hii . The distribution of f,i is useful in the applications of these test procedures. First, we discuss testing the hypotheses H,, , Hs2 ,..., HP,,-, and H simultaneously against the alternatives A,, , A,, ,..., A,,,-, and A, . In this case, we accept or reject Hi+l,i (i = l,..., p - 1) according as
where P[f<+l,i
<
a,,;
i
The joint distribution of fil ,..., a,, are not yet tabulated. But p[fi+l,i
<
%rr
=
I ye.-)
-
1
H] = (1 - a).
)
was given
f,-l,,
, i =
p
P
I,...,
-
I]
in [l5]
>
p[fp,
but the critical
<
+J.
(6.1) values
(6.2)
So, an upper bound on a,, can be obtained using the tables of the distribution of fQl. Next, consider testing the hypotheses Hgl ,..., HP,+, , and H simultaneously against the alternatives A,, ,..., A,,,-, and A, simultaneously. In this case, we accept or reject HDi according as
for practical purposes, we may choose (say) a2= . Then, u2a is given by P[fm
<
cza
the
I
critical
values
HI = (1 - a).
uIpei to be equal to
(6.3)
278
KRISHNAIAH
AND
SCHUURMANN
The total hypothesis H is accepted or rejected according as f,r 5 Us . If we test the hypotheses Hcj (; > j) and H simultaneously against Aij and A, , we accept or reject Hij according as
where
(6.4) The total hypothesis H is accepted or rejected according asfDl 5 a, . We now propose alternative procedures for testing H when some of the roots are known to be equal. Now, let pX = xi”=, Xi , Hi: hi = A, A, = u*crr [hi < A] and A, = uy=, [hi > A]. Also, let ui = ZJCy=r Zj for i = 1, 2,..., p. We will first consider the problem of testing H against A, when X, = ... = X, . In this case, we accept H if ui 3 b, and reject it otherwise
i = t,...,p -
1
where
P[ui > b,.; i = t,..., p - 1 1 H] = (1 - a).
(6.5)
If H is rejected, then we accept or reject Hi (i = t,...,p - 1) according as ui 2 bi, . If we choose the constants b, to be equal to b, (say), then the critical values b, can be obtained from the tables of the distribution of ut (under H). The hypothesis H can be also tested against A, by using any individual 4l(li + ... + I,), (i = l,..., t), as a test statistic. Next, let At = ... = X, . In this case, we accept H against A, if I(i
and reject it otherwise
d
Cia
for
i = l,..., t,
where
P[ui < ciu; i = l,..., t 1 H] = (1 - 0~).
(6.6)
If H is rejected, the hypothesis Hi (i = l,..., t) is accepted or rejected according as ui 5 ciu . If we choose the critical values to be equal to (say) c, , then c, can be obtained from Table I. The hypothesis H can be also tested against A, by using any individual li/(li + ... + Zr), (i = t,..., p), as a test statistic. hj < i Cj”=, hj], When& = ... = X, , if we wish to test H against u:Ii [p $, we may use (Zr + e.1 + I,)/(/, + 1.. + Z,) as a test statistic. The same test statistic may be also used for testing H against ul=a [p &, Xj > i Cral h,] when A, = ... = X, . One can similarly propose procedures for testing H against other restricted alternatives using Roy’s union intersection principle [ 191.
LATENT
ROOTS
OF THE
COVARIANCE
MATRIX
279
The hypothesis of the equality of the latent roots of the covariance matrix of the complex multivariate normal distribution against different alternatives can be tested by using procedures analogous to those discussed above for the real case. If we wish to test the hypothesis that /\r = ... = X, = A (where X is known) against UL, [A, < h] or lJb, [hi > X], one can use the tests based on the individual roots. Similarly, if one wishes to test H against & [Xi # d], one may use the joint density of the extreme roots. The reader is referred to [2, 3, 12, 13, 15, 211 (and some of the references in those papers) for details of these procedures and tables useful in the applications of these procedures in the real and complex cases. The procedures discussed so far for testing the hypothesis of the equality of the latent roots of the covariance matrix can be used even when the underlying populations are not real or complex multivariate normal. But the distribution problems in this case are more complicated. Procedures based on the differences (instead of the ratios) of the roots for testing the hypothesis of the equality of the roots of covariance matrix will be discussed by one of the authors in a subsequent communication. We will now discuss some applications of the procedures discussed above. samples drawn from a p-variate population Let xi ,..., xN be N independent with the cumulative distribution function (c.d.f) F(x). Also, let Fj(x), (j = 0, l,..., k), be the c.d.f. of the multivariate normal with mean vector pi and covariance matrix ~31, . Let H,, denote the hypothesis that F(x) is of the form F,-,(x) and let A, denote the alternative hypothesis that F(x) is of the form where If=, ri = 1, rrj > 0. The population covariance +-1(x) + ... + ~fl~(x) matrix of xi’s is Z = #I, + Q where 52 = xj”=, ~~j( pj - t.~)(t~,~- t.~)’ is assumed to be of rank (k - 1) and p = x:j”=, xjt+ . The problem of testing Ho is equivalent to testing for the equality of the roots of .Z when the smallest (p - k + 1) roots are equal. The alternative hypothesis A, may be restricted by imposing some conditions on the roots of Q. So, the procedures discussed before for testing H against different alternatives, may be used to test Ho. If H,, is rejected, we can use these procedures to test various subhypotheses on the roots of Z. Alternative methods of testing H,, against A, were considered by Bryant [ 11. Here we note that the sample sums of squares and cross-products (SP) matrix is distributed as the central Wishart matrix with N - I degrees of freedom when Ho is true. If A, is true, the SP matrix is distributed (see [l]) as a linear combination of the non-central Wishart matrices. The problem of testing H,, against A, is useful in the area of signal detection (see [4]). Next, let C = c2[( 1 - p) I + pee’], where e’ = (I,..., 1) and .Z is the covariance matrix of the multivariate normal distribution. If p < 0, then h, = ... = h,_i = u2[1 + (p - l)p] and &, = u2(l - p), where /\r < ...
280
KRISHNAIAH
AND
SCHUURMANN
are the eigenvalues of L’. Here, the hypothesis of p = 0 is equivalent to the hypothesis that h, = ... = h, . So, the procedures discussed before may be used for testing the hypothesis that p = 0. Next, let us assume that p > 0. Then X, = a”(1 - p) and ha = ... = h, = a*[1 + (p - I)p]. In this case, the hypothesis of p = 0 may be tested using the procedures discussed before. Consider the following model considered by Gollob [6] and Mandel [17]. yij = P + R, + Cj + ; hwUji
+ ~3 ,
i = 1).. . , s, j = 1,. . ., t,
Z=l
where yii denotes the observed value in ith row andjth column, K < min(s, t) - 1, ‘& Ri = & Ci = cb, uil = xi=, zljl = 0 and h, > ... > /\1 >, 0. Also, xi=, Z& = & Z$ = 1 and XI=, zliluil’ = Ci=, vilvjl’ = 0 for E # I’. In addition, cij are distributed independently as normal variates with zero means and variances one. Now, let W = K,Y (I” - f Js) Y’Kt’, where Y = ( yij), Js is a matrix of order s x s whose elements are one and K, is a matrix of order (t - 1) x t such that KtKt’ = I,-, and K,‘K, = It - (l/t) Jt . Without loss of generality, let us assume that t < s. Then, it is known (e.g., see [26]) that W is a Wishart matrix with (s - 1) degrees of freedom and E(W/(s - 1)) = Z* = u*lt-l + 52 where Sz = (I/(s - 1)) K,lT’K,’ , r = (vij) and vij = ‘&=r X,uilvjl . The problem of testing the hypothesis that h, = ~1.= X, = 0 is equivalent to the problem of testing the hypothesis that the eigenvalues of Q are equal since Q is not of full rank. So, the methods discussed in this section can be used to test the hypothesis that X, = *.. = h, = 0 against various alternatives; when this hypothesis is rejected, we can also draw inference as to which of the hi’s are significantly different from zero. Here, we note that it is known that the K largest eigenvalues of W are the least square estimates of h, ,..., h, . The problem of testing the hypotheses Ai = 0 was considered by Gollob [6], Mandell [17] and Yochmowitz [25] from other points of view. Next, let X’(t) = (X1(t),..., XJt)) be a p-dimensional stationary, Gaussian time series with zero means and covariance matrix R(s) = (Rij(s)) and spectral density matrix F(w) = (fii(w)) where
&j(s) = W&(t) Xi(t + 41, fij(w) = &
2 exp(--isw) Rij(s). *=--m
LATENT
ROOTS
OF THE
COVARIANCE
MATRIX
281
Let p1 < ... < pD (pl > 0, ~1~< 00) be the eigenvalues of F(w) and let us assume that
The observed time series consists of a single record of length T. A well-known estimate of F(w) is p(w) = (fij(w)), where Z* denotes the complex conjugate of 2, fidw) =
(2n :
f Iii (w + -g, 1) m=--n
and n is a suitably chosen integer. (For a review of the literature on multiple time series, the reader is referred to Parzen [18].) It is known (see [7, 231) that (2n + 1) E(w) is approximately distributed as complex Wishart matrix with (2n+ 1) degrees of freedom and E(F(w)) = F(w). So, we can test the hypothesis that F(w) = ~1~ against different alternatives by using procedures analogous to those discussed in this section for testing the hypothesis that the eigenvalues by the covariance matrix of the multivariate normal are equal.
REFERENCES [I]
P. G. (1971). Tests and estimates of the number of components in a mixture. Rept, No. 27, Department of Statistics, Stanford University. 123 CLEMM, D. S., CHATTOPADHYAY, A. K., AND KRISHNAIAH, P. R. (1973). Upper percentage points of the individual roots of the Wishart matrix. Sankhya Ser. B 35 325-338. [3] CLEMM, D. S., KRISHNAIAH, P. R., AND WAIKAR, V. B. (1971). Tables for the extreme roots of the Wishart matrix. J. Statist. Co?@. Simul. 2 65-92. [4] COOPER, D. B. AND COOPER, P. W. (1964). Nonsupervised adaptive signal detection and pattern recognition. Information and Control 7 416-444. [5] DAVIS, A. W. (1972). On the ratios of the individual latent roots to the trace of a Wishart matrix. J. Multivariate Anal. 2 4404l3. 61 GOLLOB, H. F. (1968). A statistical model which combines features of factor analytic and analysis of variance techniques. Psychometrika 33 73-116. [7] GOODMAN, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution. Ann. Math. Statist. 34 152-177. [8] JAMES, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 475-501. BRYANT,
Tech.
282 [9]
KRISHNAIAH
[ll]
[12]
[13]
[14] [I51
[16]
[17]
[18] [19] [20]
[21] [22] [23] [24]
[25] [26]
SCHUURMANN
C. G. (1965). Classical statistical analysis based on a certain multivariate Gaussian distribution. Ann. Math. Statist. 36 98-114. KRISHNAIAH, P. R. (1973). On the exact distributions of the test statistics based on the eigenvalues of complex random matrices. ARL 73-0178, Aerospace Research Laboratories, Wright-Patterson AFB, Ohio; (also in Proceedings of the 39th Session of the International Statistical Institute held at Vienna). KRISHNAIAH, P. R. AND CHANG, T. C. (1972). On the exact distributions of the extreme roots of the Wishart and MANOVA matrices. 1. Multivariate Anal. 1 108-117. KRISHNAIAH, P. R. AND SCHUURMANN, F. J. (1974). On the exact distributions of the individual roots of the complex Wishart and multivariate beta matrices. ARL 74-0063. Aerospace Research Laboratories, Wright-Patterson AFB, Ohio. KRISNAIAH, P. R., AND SCHUURMAN, F. J. (1974). On the exact joint distributions of the extreme roots of the complex Wishart and multivariate beta matrices. ARL 74-0111. Aerospace Research Laboratories, Wright-Patterson AFB, Ohio. KRISHNAIAH, P. R. AND WAIKAR, V. B. (1971). Exact distributions of any few ordered roots of a class of random matrices. /. Multivariate Anal. 1 308-315. KRISHNAIAH, P. R. AND WAIKAR, V. B. (1971). S’ imultaneous tests for equality of latent roots against certain alternatives-I. Ann. Inst. Statist. Math. 23 451-468; (also ARL Tech. Report 69-0119, 1969). KRISHNAIAH, P. R. AND WAIKAR, V. B. (1972). Simultaneous tests for equality of latent roots against certain alternatives-II. Ann. Inst. Statist. Math. 24 81-85; (also ARL Tech. Report 69-0178, 1969). MANDEL, J. (1969). The partitioning of interaction in analysis of variance. J. Res. Nut. Bur. Stand. 73B 309-328. PARZEN, E. (1969). Multiple time series modeling. In Multiwuriute Analysis--II (P. R. Krishnaiah, Ed.). Academic Press. ROY, S. N. (1957). Some aspects of multiwuriute analysis. Wiley, New York. SCHUURMANN, F. J., KRISHNAIAH, P. R., AND CHATTOPADHYAY, A. K. (1973). On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix. J. Multivariate Anal. 3 445-453. SCHUURMANN, F. J. AND WAIKAR, V. B. (1974). Upper percentage points of the individual roots of the complex Wishart matrix. (to appear) Sunkhyu Ser. B 36. SUGIYAMA, T. (1970). Joint distribution of the extreme roots of a covariance matrix. Ann. Math. Statist. 41 655-657. WAHBA, G. (1968). On the distribution of some statistics useful in the analysis of jointly stationary time series. Ann. Math. Statist. 39 1849-1862. WAIKAR, V. B. AND SCHUURMANN, F. J. (1973). Exact joint density of the largest and smallest roots of the Wishart and MANOVA matrices. Utilitus Muthemuticu 4 253-260. YOCHMOWITZ, M. (1974). A note on partitioning the interaction in a two-way model with no replication. ZMS Bull. 3 80. JOHNSON, D. E. AND GRAYBILL, F. A. (1972). An analysis of a two-way model with interaction and no replication. J. Amer. Statist. Assoc. 67 862-868. KHATRI,
complex
[lo]
AND