Evaluation of the Probability Content as an Infinite Linear Combination of Wishart Distributions

A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 152, 2000

H e le n a T. J e le n k o w s k a *

E V A L U A T IO N O F T H E P R O B A B IL IT Y C O N T E N T A S AN IN F IN IT E L IN E A R C O M B IN A T IO N O F W IS IIA R T D IS T R IB U T IO N S

Abstract. The distribution function of the homogeneous generalized quadratic form is represented as an infinite linear combination of the central Wishart distribution functions. The Probability content o f the ellipsoid is expressed as an infinite linear combination of the probability contents of spheres, under a central spherical multivariate normal distributions with unit variance, covariance matrix.

1. INTRODUCTION

Let Xj, x 2, ..., x„ be p-dim ensional ra n d o m vectors. T h en a generalized hom ogeneous q u a d ra tic form is defined as

П

£ w ; = X ' AX, (1)

i = l

w here X' is p x n ran d o m m atrix w hose colum ns are Xj, x„ an d A is a real diagonal n x n m atrix o f co n stan ts w hose d iagon al elem ents are d enoted by a u an. We shall assum e fo r convenience, w ith o u t loss o f generality, th a t 0 < a t ^ a 2 ^ . . . ^ an. W hen p = 1, X'AX reduces to a single

n hom og eneous q u a d ra tic form and it is equal to £ atx f .

i= i

F o r such single hom ogeneous an d n on-hom o gen eo us q u a d ra tic fu n ction s o f n orm al variables R u b e n (1962) expressed th e d istrib u tio n fu n ctio n as an infinite linear co m b in atio n o f chi-square d istrib u tio n fu n ctio n s with a rb itra ry scale param eter. T his result was used by R a j a g o p a l a n and

B r o e m e l i n g (1983) to find as approxim ations to the posterior distributions o f variances co m p o n en ts in u nivariate m ixed linear m odel. I о solve the abo ve problem in m u ltiv ariate case (see J e l e n k o w s k a , P r e s s , 1995) we need the results o f this paper.

In this p ap e r we shall find th e distrib u tio n fun ction o f generalized hom og eneous non-negative q u a d ra tic form o f a finite nu m b er o f correlated norm al ran d o m variables. T his d istrib u tio n will be expressed as an infinite linear co m b in atio n o f W ishart d istrib u tio n functions with a rb itra ry scale param eter.

2. DEFINITIONS AND PRELIMINARY LEMMAS

L et M be a m a trix w hose row s are p'u ц'„ w here p t = E (x,),

Л x p p X 1

i = l , ..., n and cov(Xj, Xj) = £ . A ssum e

x, ~ N ( / / „ £ ) , i = 1, n, l > 0 and sym m etric (2) p x 1

T hen

X ~ N ( M , I „ ® L ) . n X p

T h e d istrib u tio n (2) m ay be standardized by th e tran sfo rm atio n u, = E 2( x i - n i). T h a t is u , ~ N (0, I p) a nd U ~ N (0, I „ ® l p). Let K = {U: U 'A U ^ T } , T > 0, p x p , sym m etric m atrix and

I f U and T arc sym m etric m atrices, T > U m eans th a t T — U is n on n eg ativ e definite. O n replacing U = Л 2Z in (3), we o btain

H K;A(T ) = (2 л )~51Л I Í c x p i - ^ tr Z 'A - 1Z JdZ (4)

w here

R* = {Z: Z 'Z < T } .

Let Fn( ) d en o te the non sin g u lar p-dim cnsional W ish art d istrib u tio n w ith scale m atrix Ip and n degrees o f freedom , p < n , i.e.

J = i J

W ) = P

so th at

F J J ) = y _1X |T |^ " " ‘,cxp f - 2trT ) ííT’ T > 0 (5) T

an d F„(T) = 0 otherw ise, w here у is a num erical co n stan t defined as "P tíJ Ľ i) P / « 4 - 1 _ A

-

22 * д р Н

-T h en

F J J = T ) (6)

w here I is the un it n x n dim ensional m atrix.

П

W e shall show , th a t th e d istrib u tio n fu n c tio n o f £ fljUjUj m a y be i= i

expressed as a linear com b in atio n o f infinitely m an y W ish art d istrib u tio n functio ns, i.e.

Hn,A ( T = 1 с Д +2/ - т ) (7)

)=0 \a>

w here Cj = Cj„:A(cu), and со is an a rb itra ry positive c o n sta n t. G e n eratin g fu nctions for the coefficients Cj will be derived.

N ow we shall define the norm o f m atrix U as

IUI! = _{I Ui«'i}

/ = i (S)

Let Ľ = ( I, , I„ ) be a m atrix fo r th a t Ľ L = I (o rth o g o n a l m atrix ). p x n p x l p x l

I f L is a uniform ly d istrib u ted on Q, E<D(L) will be w ritten as M - o p e ra to r 00

Af<D(L). I f £<1>j(L) converges uniform ly on П, we note th at j =о

M £ Ф/ L ) = £ М Ф/ L ) (9)

j=0 J =0

N ext we ad o p t the convention th a t МФ( Ь) will be w ritten as М Ф and ЕФ(1!) as ЕФ. T h u s the arg u m en t o f Ф will be L if the expectation o p e ra to r is M and U if the expectation o p e ra to r is E. W e shall also define th a t L is induced by U if L = U ||U||_1 fo r U ^ O , we say th a t U has centered spherical d istrib u tio n if the d istrib u tio n o f P U is the sam e as th a t o f U for every o rth o g o n a l m a trix P . Im m ediately from th e ab o v e d e fin itio n it follow s th at: I f U has a centered spherical d istrib u tio n and L is induced by U, then L and || U || are independent. L is uniform ly distrib u ted o n Q.

Lem m a 1. I f U has a centered spherical d istrib u tio n , L is induced by U, Ф (и ) is a generalized hom ogeneous q u ad ratic form o f degree к and £ II Ф и II < o o , then

£ Ф (и ) = £ D U II кМФ( Ь) (10)

P roof. U sing prop erty o f hom ogeneous fun ctio n an d independence o f L and IIUII we have

Е Ф (и ) = Е[Ф( IIU II L)] = E[ IIU II кФ(Ц] = E[ || U || к]ЕФ(Ь).

ЕФ(Е) can be replaced by МФ(1.) since L is uniform ly distrib u ted on Q. I t yields the result.

L em m a 2. I f th e m a trix U: n x p has a m u ltiv a ria te s ta n d a rd iz e d spherical n o rm al d istrib u tio n then

M Ф =

- I P f ] ■) Proof. T h e density function o f U is p(U) = (2k) 2 ex p j - ^ t r U U >. Let v = £ u iU; 1=1 T hen where £ | | U | | ‘ = £ | V | ‘ = C J | V | 5 (‘ +"~p" 1)e x p ( - J t r v W k>o (. 2 j eLizA) p / „ ± i _ r\ •

n r ( 5 ± H }

3. MAIN RESULTS

In this section we prove the follow ing fu n dam ental theorem :

Theorem 1. T h e d istrib u tio n function o f the h om ogeneous generalized q u a d ra tic form (3) is represented as an infinite linear co m b in atio n o f central W ish art d istrib u tio n functions, i.e.

where at is an a rb itra ry positive co n stan t,

2 +J - 1

" 2 IAI 2E [ - ( t r Q ) ']

(13)

and

P roof. Let

Z = A2 X and Z = IIZ IIL.

D e n o tin g by

Q = Q(L) = L ' (a- 1 - ^ L (15)

we can expand exp

j

^

2|

exp { - 1 t r ö IIZ II21 = £ ( - 1)"( - t r Q Ý II Z m/m! (16)

I Z J m = 0

U sing (9) we can write

M ^ex p ^ - 2 t r ß ^ IIZ II2 J = £ ( - 1 )"M[ - ( t r o f i ] IIZ II “ /m!

By sym m etry fo r odd m

M ^ - ( t r Q ) i J = 0 m = 1, 3, ... an d (16) reduces to

M ^ exp ( - ^ t r ö ^ ) | | Z | | 2J ^ £ M [ - (trfiV ] IIZ II2J/(2j')! (17) N ext, using L em m a 2 wc obtain

(

)

P u ttin g (18) in (14)

©

T h e result follows by noting th at

r n i z r - ' - =xP { - l , r > Z ' Z ^ = Í J * ^ ^ . r ( ^ > ) F „ 2/ ( Í T ) .

T heorem 1 represents the d istrib u tio n function o f the ho m ogeneous generalized q u ad ratic form in term s o f central W ishart d istrib u tio n functions.

T he scries in (12) converges uniform ly on every finite m atrix space o f T for each t ö >0. In A d d itio n to the fo rm u la (13) for Cj o f T heo rem 1 explicit form ulae is expressed as the expectation o f a certain h om ogeneous function o f degree 2j in independent standardized n o rm al variables x ,, ..., x„.

RKKKRKNCICS

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Analysis in Statistics and Econometrics: Essays in Honor o f Arnold Zellner, John Willey

and Sons, Inc. New York, 311-321.

R a j a g o p a l a n M., B r o e m e l i n g L. (1983), Bayesian Inference for the Variance Components

o f the GeneraI M ixed Models, “ Communications in Statistics” , 12 (6), 701-724.

R u b e n H. (1962), Probability Contents of Regions under Spherical Normal Distributions IV:

the Distribution oj Homogeneous and Non-homogeneous Quadratic Function o f Normal Variables, Annals of Math. Statistics, 33, 542-570.