Asymptotic Expansions of the Distributions of Test ... - Project Euclid

J . Sci. HIROSHIMA UNIV. SER.

A-I

34 (1970), 73-144

Asymptotic Expansions of the Distributions of Test Statistics in Multivarίate Analysis Yasunori

FUJIKOSHI

(Received, April 10, 1970)

Contents 0. PART I:

1. 2. 3. 4. PART I I :

5. 6. 7. PART I I I :

8. 9. PART IV:

10. 11. 12.

Page 73

Introduction SOME USEFUL FORMULAS

75

Preliminaries Formulas for zonal polynomials Formulas for Laplace and inverse Laplace transforms of some functions of matrix argument Formulas for Laguerre polynomials

75 78 82 88

MULTIVARIATE LINEAR HYPOTHESIS

91

Asymptotic expansions of the distributions of the Pillai's criterion New derivation of an asymptotic expansion of the non-null distribution of the Hotelling's Γ§ statistic Numerical results of the powers

91

TESTS O F INDEPENDENCE

109

Asymptotic non-null distributions of the Pillai's criterion under local alternatives Asymptotic non-null distributions of the Hotelling's criterion under local alternatives O T H E R ASYMPTOTIC DISTRIBUTIONS

Calculations of

2

tr(Λd) \I— 2

APPENDIX II. Calculations of tr d 1 T — φL

109 115 120

The determinant of a non-central Wishart matrix The trace of a non-central Wishart matrix The modified LR criterion for equality of mean vectors and covariance matrices under special alternatives

APPENDIX I.

99 106

, etc. in the proof of Lemma 3 2 |

etc

APPENDIX III. Tables of the upper 5 and 1 % points of the Pillai's criterion

120 122 123 126 131 135

0. Introduction. Non-null distributions of some statistics in multivariate analysis have been expressed by series of zonal polynomials due to James Q19], especially in terms of hypergeometric functions of matrix argument (Herz [_1V\> Constantine [_6J and James C21J). Such examples have been summarized in James Q21], C22]. However the exact distributions of many test statistics are not yet available for the general values of parameters, and the almost all results obtained in some special cases are very complicated. Therefore, the asymptotic approximations for the distributions are very important.

74

Yasunori FUJIKOSHI

In this paper asymptotic expansions of the distributions of test statistics in multivariate analysis are obtained by inverting asymptotic formulas of the characteristic functions or the moment generating functions expressed in terms of the hypergeometric functions with matrix argument. Based on our formulas, an atempt is also made to compare the powers of (i) likelihood ratio ( = LR) criterion, (ii) Hotelling's criterion, and (iii) Pillai's criterion for a multivariate linear hypothesis (Wilks [42], Hotelling [13] and Lawley [24], Pillai [28]). Some statistics have been already treated by our method of asymptotic expansions. They are the generalized variance in Fujikoshi [9] and the LR criteria in multivariate analysis in Sugiura and Fujikoshi [39] and Sugiura [40]. We showed that formulas for weighted sums of zonal polynomials played an important role in our method. To extend the usefulness of our method, additional formulas are required. In part I we derive new formulas for weighted sums of zonal polynomials and the generalized Laguerre polynomials (due to Constantine [7]), and also formulas for Laplace and inverse Laplace transforms of some functions of matrix argument, which yield the asymptotic expressions of the characteristic functions. Part II deals with the multivariate linear hypothesis. For the Pillai's criterion, certain approximations have been suggested in the null case by Pillai [27], Pillai and Mijares [29], and the limiting distribution has been investigated by Ogawa [26]. In Section 5 we obtain asymptotic expansions of the distributions of the Pillai's criterion both under hypothesis and alternatives up to order 7V~2, where iV" denotes the sample size, by using the formulas for weighted sums of zonal polynomials given in Section 2. The tables of the upper 5 and \% points of the criterion based on our asymptotic expressions are given in Appendix III. Asymptotic expansion of the distribution of the Hotelling's T\ statistic has been investigated by Ito [15], [16] and Siotani [37], [38]. Recently Siotani [38] has extended his result in the nonnull case up to order N~2. In Section 6 we give two other methods for obtaining his formula, using the results due to Hsu [14] and Constantine [7], respectively. The numerical comparisons among the powers of three test criteria ((i), (ii), (iii)) are made in Section 7. We note that the observations made by Pillai and Jayachandran [30] in the case of p { — dimension of varia t e s ) ^ are also valid forp = 3, when the sample size is moderately large. In Part III, we investigate asymptotic non-null distributions of the Pillai's criterion and the Hotelling's criterion for testing the hypothesis of independence between two sets of variates under sequence of alternatives converging to the null hypothesis with rate of convergence iV"~7(7\>0). This sequence of alternatives has been considered in testing problems for covariance matrix by Sugiura [40]. By utilizing a close relationship between the tests of the multivariate linear hypothesis and the hypothesis of independence between two sets of variates, we derive asymptotic expressions of the

Asymptotic Expansions of the Distributions of Test Statistics in Multivariate Analysis

75

distributions of the Pillai's criterion and the Hotelling's criterion for this problem in the case of r = l in Sections 8 and 9, respectively. The distributions of the determinant and the trace of a non-ncentral Wishart matrix have been studied by Bagai [4], Hayakawa [10], etc. However, the exact distributions which they have obtained in some special cases are too complicated for numerical computation. In Sections 10 and 11 we investigate asymptotic distributions of the statistics for large n ( = degrees of freedom of the non-central Wishart matrix) under the assumption that the non-centrality matrix Ω = n8Θ(δ'>0) and Θ does not depend on n. In Fujikoshi Q9] we obtained an asymptotic expansion of the determinant of the non-central Wishart matrix in the case that Ω is a fixed constant matrix, i. e., in the case of δ~0, by using Lemma 2 in Section 2. In the same way we can derive it in the case of δ = -j- up to order n"~2 in Section 10. In Section 11, by using the explicit expression of the characteristic function of the non-central Wishart matrix due to Anderson [1] we obtain asymptotic expansions of the distributions of the trace of the non-central Wishart matrix __ 3

up to order n 2, when δ = 0 and 1, without using the formulas in part I. In Section 12 we obtain asymptotic expansions of the non-null distribution of the modified LR criterion for equality of mean vectors and covariance matrices under the restricted alternatives such as equality of all the covariance matrices. PART I.

SOME USEFUL FORMULAS

1. Preliminaries. We list some necessary results on zonal polynomials and others which will be used frequently in this paper. The hypergeometric function of matrix argument is defined by Constantine [6] as (1.1)

rFs(au-

-9 ar; bu

= Σ Σ {(aι)κ

->bs;Z) -(ar)Aί>ι)«

--(bs)κ}Cκ(Z)/kl,

where au -,ar, bι,.-,bs a r e real or complex constants, κ= {ku &2, , kp) denotes a partition of t h e integer k such t h a t kλ + k2H \-kp = k and & i > & 2 > •• Further (1.2)

(α).= Π ( α - ( α a=ι

and the symbol Σ denotes summing over all partitions for fixed k.

Also,

the function CK(Z) is called a zonal polynomial of thepxp symmetric matrix Z corresponding to K and it is a symmetric homogeneous polynomial of degree

Yasunori FUJIKOSHI

76

k in the p characteristic roots of Z. The detailed discussion may be found in Constantine [6] and James [19], [20], [21]. Tables for zonal polynomials have been given by James [21] up to order 6. For & = 1, 2, 3 and 4, (1.3)

) = trZ, C(2)(Z)

1 '3

11

C(3)(Z)

1 15 ){Z)

105 -(21

2

(tr Z) 2

2

trZ 8

6 9

-18

, 5 -15

10

9

' ;

2

/'

(tr Z) 3

2

(tr Z)tr Z j ,

"'•• t r Z

3

ί

(tr Z) 4

1

12

12

32

48

20

100

-40

80

-160

(tr Z) 2 tr Z 2

14

28

98

-112

-28

(tr Z 2 ) 2

56

-56

-112

-112

224

(tr Z)tr Z 3

14

-84

42

112

-84

tr Z 4

The fundamental property of zonal polynomials is the average over the orthogonal group O(p), given by (1.4)

O(P)

where / is the identity matrix of order p and dju (H) is the invariant measure on the orthogonal group O(p), so normalized that the measure of the whole group is unity. Special cases of the hypergeometric function of matrix argument are (1.5)

0

F 0 (Z)=Σ

l = etr Z,

z) = Σ Σ

(1.6)

k

)

where the last formula (1.6) holds when all the absolute values of the characteristic roots of Z are less than one. The following recurrence relations for the hypergeometric function defined in (1.1) are due to Constantine [6]: (1.7)

S>Ό

{etr(-S)}\S\a-(p+1)l\Fs(au...,ar;bu...,bs;ST)dS

Asymptotic Expansions of the Distributions of Test Statistics in Multivariate Analysis = r + iFs(aiy, (1 8 )

ar, a; δ i ,

, bs; T ) ,

{etϊ

T)\T\-KFs(au-,

= r^ 7 s + i ( « i 5

5 a,r\ δ i ,

Wβ^~\^--xa>o

77

aribu-, bs; T^S)dT

, ό s , b\ 5 ) ,

which may be regarded a generalization of Laplace and inverse Laplace transformation. The integral of the last formula (1.8) is taken over all T(p xp) = Xo + ίY for fixed positive definite matrix Xo and arbitrary real symmetric matrix Y. The function Γp(t) is defined by Γ,(ί) = τ r ^ " 1 ) / 4 Π Γ ( ί - ( α - l ) / 2 ) .

(1.9)

The following formulas for zonal polynomials are also obtained by Constantine [6]. (1.10)

( J S>0

where the first formula (1.10) holds for any symmetric matrix Z whose real part (=3t(Z)) is positive definite and any symmetric matrix T for 3l(ί)> (/> —1)/2. The last formula (1.11) holds for any positive definite matrix 5. The Laguerre polynomial of matrix argument is defined by Constantine as follows: ( U 2 )

o

where γ>—l. polynomials: (1.13)

ίΊ(r+4-

He obtained the following generating function for Laguerre

Σ ΣLl(Ω)Cκ(Z)/{k\CXI)} =

(p+1)

\I-Z\-^

2

kp} of k withkι + k2-\ hkp = k and &i>& 2 > ••>&£>(). Putting

LEMMA

κ={ku (2.1)

= Σ ka ι

then the following differential relations hold: (2.2)

ai

(ic)Cκ ( Z ) = t r (Λd)2 Cκ (Σ)

(2.3) Σ=Λ

where d denote the matrix of differential

operators having 00r

s

as its (r, 5) element for a symmetric matrix Σ — ((7rs) with Kronecker's delta δrs and A — diag (λu λ2y^ λp) is a diagonal matrix with p characteristic roots of Z as its non-zero elements. It is sufficient to prove that the formulas (2.2) and (2.3) hold for any positive definite matrix Z. From (1.10) we have the following asymptotic formula for large n with any positive definite matrix Z: PROOF.

(2.4)

{ Γ ^ ^ l Z l t p J ^ {etr(-Z-15)> \ S\^-p^l2Cκ^S^j dS


79

The left hand side can be expanded asymptotically as follows, by using the method of a matrix of differential operators (see Ito [15], James [23], Siotani [37], Sugiura and Fujikoshi [39], etc. for the method.) (2.5)

Comparing the coefficients of each term of orders n'1 and n~2 in the last equations of (2.4) and (2.5), we see that the formulas (2.2) and (2.3) are true. The following seven lemmas are fundamental for the asymptotic expansions. Lemma 2 will be used in Sections 10 and 12. Lemma 3 will be used in derivation of the Pillai's criteria for multivariate linear hypothesis and independence. LEMMA

2 (Sugiura and Fujikoshi [39]).

The following identities hold:

(2.6)

ΣΣCK (Z)/(k -1)1 = (tr zyetr Z,

(2.7)

ΣΣCK (Z)m (jc)/k \ = (trZ2) etr Z,

(2.8)

Σ Σ^(Z)α!(Λ:)/(A;-l)!={2trZ 2 + (tr Z) tr Z 2 }etr Z,

(2.9)

Σ Σ Cκ(Z)a1(fc)2/k != {(tr Z)2-f- tr Z 2 + 4tr Z 3 + (tr Z 2 ) 2 } etr Z,

(2.10)

Σ ΣCκ(Z)a2(/c)/kl={tτ Z + 3 ( t r Z) + 3tr Z + 4 t r Z }etr Z,

2

4

2

2

3

3

2

3

= {12tr Z +8(tr Z )tr Z + (tr Z ) }etr Z where a3(fc) is defined by (2.12)

a3(ιc)=Σka{2ki-4: k= l

PROOF. For a proof of the formulas (2.6)^(2.10), see Fujikoshi [9], Sugiura and Fujikoshi [39]. In the following we shall prove the formula (2.11). From (1.6) we can write

(2.13)

\I-n-1Z\-»=Σ k

Σ(n)κCκ(Z)/{nkk*} (

80

Yasunori FUJIKOSHI

= Σ Σ \l + k=o (*) L

(2n)-1a1(ιc)+(24n2y1{Sa1(κ)2-a2(κ

which holds for any symmetric matrix Z with large n. The left hand side can be expanded asymptotically as in (2.15) by the well known formula, (2.14). - log I I-n~ιZ

(2.14)

\ = Σa'

\I-n-1Z\-n

(2.15)

= (etr Z)Γl + ( 2 « ) - 1 t r Z 2 + (24re2)"1{8tr Z 3 + 3(tr Z 2 ) 2 } + (48re3)"1 {12tr Z4+ 8(tr Z 2 )tr Z 3 + (tr Z 2 ) 3 } Comparing the coefficients of each term of order n~3 in (2.13) and (2.15), we obtain the formula (2.11). LEMMA 3. Let Z be any symmetric matrix such that all the absolute values of the characteristic roots are less than one and put F=Z(I—Z)~1. Then the following identities hold:

(2.16) (2.17)

?)/(A-2)! = = b{δ(tr vy+ tr F 2 }

k = 2 (*) oo

(2.18)

/ , / | V^A^/ίV & = 0 (*)

4

6

{(tr Γ) + (26 + 1) t i • F 2 } |/-z|-

?)α 1 (/c)/(A —•D!

= {{2(tr F )

3

+ b(tr V) + (2δ (2.20)

-b

?K (*)/*! =

oo

(2.19)

1/

2

,

+ 2:(2!ό + l

2: -b 2)( tr Γ ) t r F + 2(2ό + :L)tr V*}\I-Z\ j

2 + δ +

r)αiW2Ah

2

=

τ {2(26 + 1) (tr

+ 4(tr V)3 +12(26+ l)(trΓ)tr Γ 2 + 8/ O Ϊ 2 ! + 2(262 + δ• + 2)(tr Γ)t5r

+ 8(2i + l)(tr F)tr Γ

Vf ; +2(26 + 3)tr F^ _|_ ς

3

!δ- l-2)tic F + δ(tr F) 2

-δ + 2)(tr F )

+ 5)tr F 4 }\i-z\-\

2

4


81

Σ Σ ( a C ( 2 ) « 2 W A ! = | { 2 t r F + 3(2ό + l)(tr V)2

(2.21)

ώ

k=0 («)

2

+ 3(2ό + 3)tr Γ + 2(tr Γ) 3 + 6(2ό + l)(tr V)tτ V2 + 4(2ό2 + 3ό + 2)tr

V3}\I-Z\'K

In (1.6) replacing Z by xZ, we have

PROOF.

Σ Σ χk(b),CXZ)/k \ = \I-xZ\ -»,

(2.22)

which holds for any x such that | x | < 1 . Differentiating (2.22) with respect to x by -^- I U(x) I = I U(x) I tr U(χYι ~- U(x), we have the formulas (2.16) (/ X

(s X

and (2.17). Multiplying both sides of (2.2) by (b)κ/k\ and using the formula (1.6), we obtain Σ Σ(b)κCκ(Z)aι(/c)/k\=tr(Λd)2\I-Σ\

(2.23)

-b

After some calculations, we see that the right hand side of (2.23) is equal to the right hand side of (2.18) (see Appendix I). The formula (2.19) follows immediately by replacing Z by xZ in (2.18) and differentiating it with respect to x. Multiplying both sides of (2.2) by (b)κaι(fc)/kl and using the third formula (2.18), we obtain (2.24)

Σ Σ (b)j k = 0 (*)

whose right hand side is reduced to the formula (2.20) (see Appendix I). Similarly multiplying both sides of (2.3) by (b)κ/k\ and using the formula (1.6), we obtain (2.25)

Σ k

(

Σ(b)κCκ{Z)

Using the formulas (A. 21) and (A. 22) shown in Appendix I, we can write the right hand side of (2.25) as follows: | -| /| /Z- Z | -| -* *{ {88((ttr F) 3 + 24(2ό + l)(tr F)tr F 2 + 16(2ό2 + 3ό + 2)ίr FF3 + 3δ(tr Γ) 4 + 6(2ό2 + & + 2)(tr Γ) 2 tr Γ 2 + 3(2ό + l)(2ό 2 + ό + 2)(tr V2)2

82

Yasunori FUJIKOSHI

+ 24(2δ + l)(tr Γ ) t r V3 + 6(8b2 + l0b + 5)tr V4}. Combining this result with (2.16) and (2.20), we have the desired formula (2.21). 3.

Formulas for Laplace and inverse Laplace transforms of some functions

of matrix argument. We will use the following abbreviated notations: i s>o

for any symmetric matrix Z whose real part (= Xo) is positive definite, and \6.Δ)

l|_y(Z

, SJ, (y)_\ (p\r 7}\7\~~ϊ~f(7~1 Ω Θ) dZ VCUL ^ ) I ^ I J V^ 5 u0

{etτ-(I

+Ω2Z-1ΩhR}\R\(9~p~1)l2{

}dR\dZ9

and using Lemma 2, we obtain the following identities: OO

(4.10)

Σ Σ*kLl(ώ)aλ(κ)/k!

= (etr Ω)x2YLΩ[(tr R2)etr xRΓ],

(4.11)

Σ Σ xkLI(Ω)ax(ιc)2/kl= (etr i2)x 2 IL β [{(tr R)2+ tr R2 Φ 4 r t r K34-r2(ϊr R2V\PΪV \^ *±Λ bJL J.\.

r^ Λ/ yUJ. J.X. J j C l / i

rffΊ Λ/JL\ I ,

OO

+ Sx tr R2 + Ax2 tr R3} etr xR~]. Using Lemmas 4 and 5, we simplify each of the right hand sides of above expressions. For example, by Lemma 4 the right hand side of (4.10) can be written as follows: (4.10)

\x

with I [ { }] defined by (3.2). Making the transformation we can write it as follows:

(4.13)

Z->Z+(l-χ)~ιΩ,

fα--)^ t r

ΩZ

~l 1—x

ι 2

. / t r ΩZ~ \ V 1—x /

(1 — x)

Applying Lemma 5 to the above expression, we obtain the right hand side of (4.4). Similarly the right hand sides of (4.11) and (4.12) also imply (4.6) and (4.7), respectively.


PART II.

5.

91

MULTIVARIATE LINEAR HYPOTHESIS

Asymptotic expansions of the distributions of the Pillai's criterion

5.1. The moment generating functions of the criterion. The multivariate linear hypothesis model has been discussed by many authors (e.g., Anderson [2], Das Gupta, Anderson and Mudholkar [8], Roy [35], Seber [36], etc.). The following canonical form is well known: Let each column vector of pxN

matrix X' = (X[(pXq), X'2(px(N-s)\

X'3(px(s-q)))

with q-variate normal distribution with the common covariance matrix Σ and expectations given by = M(qxp), E[X 2 ] = 0((JV-s) xP\

E[X 3 ] = Γ ( ( s - ? ) xp).

Then multivariate linear hypothesis is defined by testing the hypothesis (5.1)

H: M= 0

against alternative

where Γ is a matrix of nuisance parameters. [28]) for this problem is based on the statistic (5.2)

K: MφO, The Pillai's criterion (Pillai

V=mtr Sk(Sk + Sβy\

where m = N—s + q = n + q9 Se = X'2 X2 and Sh = X{Xi are the matrices of sums of squares and products due to error and due to the hypothesis, respectively. The matrix Se has the Wishart distribution Wp(n, Σ). The matrix Sh has the non-central Wishart distribution Wp(q, Σ, Ω), where the matrix of non-centrality parameters is given by Ω = -j~Σ *M' MΣ 2. We can easily see that in the case of q^>p under the hypothesis H the 2

J

matrix 5 = (5Λ + 5e) Sh(Sh τSe) tribution

(5.3)

2

Γ,(^±i) \rp(±yp(±.)}

has the following multivariate beta dis-

\B\«->-w\I-B\«->-w dB.

Therefore, by James [21] the moment generating fonction of V under the hypothesis H with q^>p is expressed as follows: (5.4)

MH(t)=ι

By using a well known fact that the density function of the characteristic 1 roots of Sh S7 , in the degenerate case of q ω{ mean by our notations 2ra + 3, 2^ + 3, -γωi9 respective2 ly) is excellent and is still excellent in the case when terms of O(m~ ) are neglected. Tables 1 and 2 also shows that the approximations to the upper h% points are better than those of the upper \% points. It is worthwhile to note that our asymptotic formulas (5.11) and (5.19) hold for any p, q, n such that p < ft and without any assumption on the rank of noncentrality matrix Ω. By using (5.21), the upper 5 and \% points were computed for values /> = 2(1)7, ? = 2(1)12, and Λ = 25, 30, 40, 60, 80, 100, 130, 160, 200, 250, 350, 500. These results are presented in Appendix III. 6. New derivation of an asymptotic expansion of the non-null distribution of the Hotelling's Ύ% statistic

6.1. The characteristic function and the Laplace transform of the statistic. The Hotelling's criterion for testing the multivariate linear hypothesis given in (5.1) is based on the statistic Tl = n tr ShS~ι, which is called the Hotelling's Tl statistic. The exact distribution of this statistic has been studied by various authors, e.g. Hsu [14], Hotelling [13], Constantine [7], Pillai and Jayachandran [30]. However the exact distribution of Tl is available only for some particular values of/? and q or under the condition Tlo 1 •

ΓΓ

Lj5>o

Ί

±

ι

L

J

We must carry out each integral j4fi[Xtr ^) 2 ] 5 ^^Etr S 2 ], etc. in (6.11). For example, let us consider j4βC(tr 5) 2 ]. By using Lemma 4, we can write (6,13) . ^ [ ( t r Syj = rJ±)

Λ

λHp+1)l

2

(etr T)\ Γf

|(l-2ίί)J


x

i

τ~ ώ I ~*-|j? {tr((l-2iί) I-2itΩ

103

1

Γ" ^

1

Considering the transformation T-*Z= T—2it(l — 2it)~ Ω, we can simplify 2 AC(tr S) J as follows:

with the notation IQ ] in (3.2). Applying Lemma 5 to the above expression, we obtain (6.14)

The similar computation gives us (6.15) (6.15)

/J β [tr5]= (l

S)tr 5 2 ] - ( l - 2 f

2τίk

104

Yasunori FUJIKOSHI

Ω)2+ (qp2+ (q2+ q+ 20) p 3 2

+ 4(5