STOCHASTIC ML\JORIZATION OF THE LOG ... - CiteSeerX
STOCHASTIC ML\JORIZATION OF THE LOG-EIGErWAWES OF ABIVARIATE WISHART Ml\TRIX
BY MICHAEL D. PERLMAN
TECHNICAL REPORT NO. 39 SEPID1BER 1983
DEPARTMENT OF STATISTICS UNIVERSITY OF WASHINGTON SEATTLE) WASHINGTON
98195
srOCHASTIC MAJORIZATION OF THE LOG-EIGENVALUES OF A BIVARIATE WISHART MATRIX1.2 by Michael D. Perlman
Department of Statistics University of Washington Seattle, Washington 98195
ABSTRACT
Let 1 = (l1' l2) and A = (A1. A2). where A1;:f; A2 > 0 are the ordered eigenvalues of Sand:E, respectively, and S .... Wz(n, E) is a bivariate Wishart matrix. Let m = (mlo m2) and JL = (f.L1. f.L2). where 1T'/i log 4 and /-4. log Ai' It is shown that P _fm jl' B l is Schurconvex in J,l whenever B is a Schur-monotone set, i.e. [x E B, x majorizes x"] ::;> x· E B. This result implies the unbiased-
=
=
ness and power-monotonicity of a class of invariant tests for bivariate sphericity and other orthogonally invariant hypotheses.
Abbreviated Title: Log - Eigenvalues of a Wishart Matrix MIS 1960 Subject Classifications: Primary 62H15; Secondary 62H10 Key Words and Phrases: Bivariate Wishart distribution. eigenvalues, stochastic majorization, Schur function.
log-
1983 research was supported in part by National Science Foundation Grants MCS 80-02167 and MCS83-01807.
- 2-
§ 1. INTRODUCTION. Let S "'" W2 (n, 2;) be a bivariate Wishart matrix with n degree of freedom (n ~ 2) and expected value n2; (2; positive definite). We shall be concerned with the power functions of orthogonally invariant tests for invariant testing problems such as the following:
(1.1 )
Hal: 2; = crI, cr arbitrary vs. «; vs. K 2 : H o2 : 2;=! vs. K 3 : Has: 2;=! vs. K 4 : H o4 : 2;=1
arbitrary 2; arbitrary 2; - I positive definite 2; - I negative definite. 2;
Orthogonally invariant tests depend on S only through 1 == (l1> l2), where II ~ l2 (> 0) are the ordered eigenvalues of S. Because the power function of such tests depend on 2; only through A == (Ab A2), where x, ~ A2 (> 0) are the ordered eigenvalues of 2;, we may assume throughout this paper that 2; = D). == diag(Al' A2)' The notions of majorization and Schur-convexity play an important role in determining such properties as unbiasedness and power monotonicity of invariant tests. To illustrate, consider the likelihood ratio test (LRT) for testing Hal (bivariate sphericity) vs. s ; The acceptance region, can be expressed in the equivalent forms
(1.2)
! S
trS I ~ll/2
I !I
~ c
+--7
1
II +
(ll l2)
l2
I
~c .
Since
( 1.3)
trS e
where S
= (Sij) i.j=I.2,
R is the sample correlation coefficient, and ti log Sii, and since SI1> S22, and R are independent with Sii "'" AiXn 2 when 2; = D}", conditioning on R reduces the problem to the study
=
power of of based on the independent I-variates s 11 and degrees of freedom. is easy to show that (AI = ~)
t
S22
with equal density of
3
(1.4)
ItI :"
+ e"
~ c 1 o. For r
Ir ~ c l
+im21 r ~ c ~
~ 1,
the symmetric of so corresponding f.L. Other acceptance regions possibly appropriate are regions nAlATt::J,y>
VUl. O. For each r > 0, B rr]is a Schur-monotone region in 1R§ by (2.2), so the corresponding power function is Schur.convex in JL. Note that for 0 < r < 1, Brr ] is not convex. Next, we discuss a class of one-sided acceptance regions based on tr(sr) appropriate for testing H oa : J.l.l = J.l.2 = 0
For
-00
~
r ~
00
vs.
K s : J.l.l ~ J.l.2 ~ 0 .
define
note that
- l 2-T -00-
e
ma J
by continuity. The equivalent acceptance regions
+--+
s; = [m I t; ;;;; c J
are appropriate for testing Hos us. K s. For each r ~ 0 (r ~ 0) the symmetric extension of s, (B r C ) to 1R2 is convex and permutationinvariant, so by Theorem 2.4 the corresponding power function is Schur-convex (Schur-concave) in JL. [For r = 0, the distribution of To == ~11/2 depends on J.l. only through J.l.l + J.l.2 == log I:E!, so the power function corresponding to B o is trivially both Schurconvex and Schur-concave in JL.] Similarly, the equivalent regions ArC +--+ B r c complementary to A r +--+ B; are appropriate acceptance regions for testing
follows associated concave) for
r
paragraph the acceptance region
nowp,r function
Schurconvex
-8-
the equivalent forms Aos =
lilt..~nl
B oa = fm
+-"-l-
4
fl I I;
4 +1]
[log (--) - -
n
n
LJ 7T/.i I Ii 1m.,;;;;:"[( nl
~
C
I
(m.-logn) log n) - e ' + 1] ~
) Cl
log
=
Jml t: 1
tp
(7T/.i - log n)
t=l
~ cl'
where
I
x - e" + 1
rp(x)=
0
~ 0, if x ~ O.
if x
Since rp is a concave function on (-00.00). Bos is a Schur-rnonotone region in JRff and the associated power function is Schur-convex in fJ.. Similarly, the acceptance region of the LRT for H 04 vs. K 4 can be expressed as A 04 =
+-"-l-
B 04
fl I I;
lilt..~nl
4 l· [log (--) - ~ + 1] ~ c ] n
=ftm I .E 1/1(7T/.i t=l
log n)
n
~ cl'
where 1/I(x) =
I
x - eX + 1 0
if x if x
~. 0 ~ O.
Since 1/1 is concave on (-00,00), it follows as above that the power function associated with B 04 is Schurconvex in u: Other power monotonicity properties of some of the tests discussed in this section may be found in Anderson and Das Gupta (1964), Das Gupta (1969), and Das Gupta and Girl (1971).
§ 4:. CONCLUDING REMARKS. The proof expressing the eigenvalues ll. working with approach may
han>","""""
2.4 proceeded
terms of
l
diagjoint to
0U·l.lfJJ.'V
-
_._~~ 1
"'2\.\,"'1.1.\..1.
9 carry through the latter approach in the bivariate case. Theorem 2.4 can be extended from probabilities of Schurmonotone regions to expectations of Schur-convex functions: if g is a Schur-convex function on IR§ such that the expectations exist, then s; [gem)] is a Schur-convex function of IJ,. This follows by a standard approximation argument.
- 10
REFERENCES
Anderson, T. W. and Das Gupta, S. (1964). A monotonicity property of the power functions of some tests of equality of two covariance matrices. Annals of Mathematical Statistics 35 10591063. [Correction: ibid 36 (1965), 1318.] Das Gupta, S. (1969). Properties of power functions of some tests concerning dispersion matrices of multivariate normal distributions. Annals of Mathematical Statistics 40 697-701. Das Gupta, S. and Giri, N. (1971). Properties of tests concerning covariance matrices of normal distributions. Annals of Statistics 1 1222-1224. GIeser, 1. J. (1966). A note on the sphericity test. Annals of Mathematical Statistics 37 464-467. Marshall, A. W. and Olkin, 1. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York. Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. John Wiley & Sons, New York. Perlman, M. D. (1967). One-sided testing problems involving covariance matrices. Unpublished notes, Department of Statistics, Stanford University. Perlman, M. D. (1982). Stochastic comparison of eigenvalues of Wishart matrices, with applications to unbiasedness of tests for variances and covariance matrices. Invited address presented at on and Probability, of Nebraska, October 27-30, 1982.
BY MICHAEL D. PERLMAN
TECHNICAL REPORT NO. 39 SEPID1BER 1983
DEPARTMENT OF STATISTICS UNIVERSITY OF WASHINGTON SEATTLE) WASHINGTON
98195
srOCHASTIC MAJORIZATION OF THE LOG-EIGENVALUES OF A BIVARIATE WISHART MATRIX1.2 by Michael D. Perlman
Department of Statistics University of Washington Seattle, Washington 98195
ABSTRACT
Let 1 = (l1' l2) and A = (A1. A2). where A1;:f; A2 > 0 are the ordered eigenvalues of Sand:E, respectively, and S .... Wz(n, E) is a bivariate Wishart matrix. Let m = (mlo m2) and JL = (f.L1. f.L2). where 1T'/i log 4 and /-4. log Ai' It is shown that P _fm jl' B l is Schurconvex in J,l whenever B is a Schur-monotone set, i.e. [x E B, x majorizes x"] ::;> x· E B. This result implies the unbiased-
=
=
ness and power-monotonicity of a class of invariant tests for bivariate sphericity and other orthogonally invariant hypotheses.
Abbreviated Title: Log - Eigenvalues of a Wishart Matrix MIS 1960 Subject Classifications: Primary 62H15; Secondary 62H10 Key Words and Phrases: Bivariate Wishart distribution. eigenvalues, stochastic majorization, Schur function.
log-
1983 research was supported in part by National Science Foundation Grants MCS 80-02167 and MCS83-01807.
- 2-
§ 1. INTRODUCTION. Let S "'" W2 (n, 2;) be a bivariate Wishart matrix with n degree of freedom (n ~ 2) and expected value n2; (2; positive definite). We shall be concerned with the power functions of orthogonally invariant tests for invariant testing problems such as the following:
(1.1 )
Hal: 2; = crI, cr arbitrary vs. «; vs. K 2 : H o2 : 2;=! vs. K 3 : Has: 2;=! vs. K 4 : H o4 : 2;=1
arbitrary 2; arbitrary 2; - I positive definite 2; - I negative definite. 2;
Orthogonally invariant tests depend on S only through 1 == (l1> l2), where II ~ l2 (> 0) are the ordered eigenvalues of S. Because the power function of such tests depend on 2; only through A == (Ab A2), where x, ~ A2 (> 0) are the ordered eigenvalues of 2;, we may assume throughout this paper that 2; = D). == diag(Al' A2)' The notions of majorization and Schur-convexity play an important role in determining such properties as unbiasedness and power monotonicity of invariant tests. To illustrate, consider the likelihood ratio test (LRT) for testing Hal (bivariate sphericity) vs. s ; The acceptance region, can be expressed in the equivalent forms
(1.2)
! S
trS I ~ll/2
I !I
~ c
+--7
1
II +
(ll l2)
l2
I
~c .
Since
( 1.3)
trS e
where S
= (Sij) i.j=I.2,
R is the sample correlation coefficient, and ti log Sii, and since SI1> S22, and R are independent with Sii "'" AiXn 2 when 2; = D}", conditioning on R reduces the problem to the study
=
power of of based on the independent I-variates s 11 and degrees of freedom. is easy to show that (AI = ~)
t
S22
with equal density of
3
(1.4)
ItI :"
+ e"
~ c 1 o. For r
Ir ~ c l
+im21 r ~ c ~
~ 1,
the symmetric of so corresponding f.L. Other acceptance regions possibly appropriate are regions nAlATt::J,y>
VUl. O. For each r > 0, B rr]is a Schur-monotone region in 1R§ by (2.2), so the corresponding power function is Schur.convex in JL. Note that for 0 < r < 1, Brr ] is not convex. Next, we discuss a class of one-sided acceptance regions based on tr(sr) appropriate for testing H oa : J.l.l = J.l.2 = 0
For
-00
~
r ~
00
vs.
K s : J.l.l ~ J.l.2 ~ 0 .
define
note that
- l 2-T -00-
e
ma J
by continuity. The equivalent acceptance regions
+--+
s; = [m I t; ;;;; c J
are appropriate for testing Hos us. K s. For each r ~ 0 (r ~ 0) the symmetric extension of s, (B r C ) to 1R2 is convex and permutationinvariant, so by Theorem 2.4 the corresponding power function is Schur-convex (Schur-concave) in JL. [For r = 0, the distribution of To == ~11/2 depends on J.l. only through J.l.l + J.l.2 == log I:E!, so the power function corresponding to B o is trivially both Schurconvex and Schur-concave in JL.] Similarly, the equivalent regions ArC +--+ B r c complementary to A r +--+ B; are appropriate acceptance regions for testing
follows associated concave) for
r
paragraph the acceptance region
nowp,r function
Schurconvex
-8-
the equivalent forms Aos =
lilt..~nl
B oa = fm
+-"-l-
4
fl I I;
4 +1]
[log (--) - -
n
n
LJ 7T/.i I Ii 1m.,;;;;:"[( nl
~
C
I
(m.-logn) log n) - e ' + 1] ~
) Cl
log
=
Jml t: 1
tp
(7T/.i - log n)
t=l
~ cl'
where
I
x - e" + 1
rp(x)=
0
~ 0, if x ~ O.
if x
Since rp is a concave function on (-00.00). Bos is a Schur-rnonotone region in JRff and the associated power function is Schur-convex in fJ.. Similarly, the acceptance region of the LRT for H 04 vs. K 4 can be expressed as A 04 =
+-"-l-
B 04
fl I I;
lilt..~nl
4 l· [log (--) - ~ + 1] ~ c ] n
=ftm I .E 1/1(7T/.i t=l
log n)
n
~ cl'
where 1/I(x) =
I
x - eX + 1 0
if x if x
~. 0 ~ O.
Since 1/1 is concave on (-00,00), it follows as above that the power function associated with B 04 is Schurconvex in u: Other power monotonicity properties of some of the tests discussed in this section may be found in Anderson and Das Gupta (1964), Das Gupta (1969), and Das Gupta and Girl (1971).
§ 4:. CONCLUDING REMARKS. The proof expressing the eigenvalues ll. working with approach may
han>","""""
2.4 proceeded
terms of
l
diagjoint to
0U·l.lfJJ.'V
-
_._~~ 1
"'2\.\,"'1.1.\..1.
9 carry through the latter approach in the bivariate case. Theorem 2.4 can be extended from probabilities of Schurmonotone regions to expectations of Schur-convex functions: if g is a Schur-convex function on IR§ such that the expectations exist, then s; [gem)] is a Schur-convex function of IJ,. This follows by a standard approximation argument.
- 10
REFERENCES
Anderson, T. W. and Das Gupta, S. (1964). A monotonicity property of the power functions of some tests of equality of two covariance matrices. Annals of Mathematical Statistics 35 10591063. [Correction: ibid 36 (1965), 1318.] Das Gupta, S. (1969). Properties of power functions of some tests concerning dispersion matrices of multivariate normal distributions. Annals of Mathematical Statistics 40 697-701. Das Gupta, S. and Giri, N. (1971). Properties of tests concerning covariance matrices of normal distributions. Annals of Statistics 1 1222-1224. GIeser, 1. J. (1966). A note on the sphericity test. Annals of Mathematical Statistics 37 464-467. Marshall, A. W. and Olkin, 1. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York. Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. John Wiley & Sons, New York. Perlman, M. D. (1967). One-sided testing problems involving covariance matrices. Unpublished notes, Department of Statistics, Stanford University. Perlman, M. D. (1982). Stochastic comparison of eigenvalues of Wishart matrices, with applications to unbiasedness of tests for variances and covariance matrices. Invited address presented at on and Probability, of Nebraska, October 27-30, 1982.
