StaUstics & Probablhty Letters 7 (1989) 207-216 December 1988 North-Holland

M U L T I V A R I A T E D I S T R I B U T I O N S O F O R D E R k

Andrea s N. P H I L I P P O U , Demet r i s L. A N T Z O U L A K O S and Gregory A. T R I P S I A N N I S

Department of Mathematzcs, Unwerszty of Patras, Patras, Greece

Received November 1987 Revised February 1988

Abstract: Three mulUvariate dlstnbuttons of order k are introduced and studied. A multivariate negaUve blnormal dlstnbuuon of order k is derived first, by means of an urn scheme, and two hnutlng cases of it are obtained next. They are, respecuvely, a muluvanate Polsson dlstrtbutlon of order k and a multivariate loganthrmc series d~stnbut~on of the same order. The probab~hty generating functions, means variances and covanances of these distributions are obtamed, and some further genes~s schemes of them and mterrelat~onsbaps among them are also estabhshed. The present paper extends to the multivariate case the work of Phihppou (1987) on multlparameter distributions of order k. At the same ttme, several results of Akd (1985) on extended dlstnbutaons of order k are also generahzed to the multwanate case.

Keywords: multivariate dlstrtbutlons of order k, negaUve bmormal, Po~sson, loganthrmc senes, probabdlty generating functions, means, variances, covanances, genes~s schemes, interrelationships.

1. Introduction and summary

In a recent paper , Phi l ippou (1987) der ived a mu l t i pa r ame te r negat ive b inomia l d i s t r ibu t ion of o rde r k b y c o m p o u n d i n g the ex tended (or mul t ipa ramete r ) Polsson d i s t r ibu t ion of o rde r k of Aki (1985) by the g a m m a d is t r ibu t ion . He also ob ta ined a mu l t i pa rame te r logar i thmic series dlstr ibut~on of o rde r k as the zero t runca ted l imi t of the first, and es tabl i shed a few genesis schemes and in te r re la t ionsh ips for these three mu l t i pa r ame te r d i s t r ibu t ions of o rder k. In the present paper , we ex tend the work of Ph i l ippou (1987) to the mul t ivar ia te case. In Sect ion 2, we der ive a mul t ivar ia te nega twe b inomia l d i s t r ibu t ion of o rder k, b y means of an urn scheme (see Propos i t ion 2.1 and Def in i t ton 2.2), and we ob t a in two l imi t ing cases of it (see Theorems 2.3 and 2.7) which provide , respectively, a mul t iva r ia te Polsson d i s t r ibu t ion of o rder k and a mul t iva r ia te logar i thmic series d i s t r ibu t ion of the same order (see Def in i t ions 2.4 and 2.8). In Section 3, we get the p robab i l i t y genera t ing funct ions, means, var iances and covar iances of these three d is t r ibut ions , and give some fur ther genesis schemes of them and in te r re la t ionsh ips among them (see

Propos i t ions 3 .5 -3 .7-and Theorems 3.1, 3.8 and 3.10). The present p a p e r general izes several results on mul t ivar ia te d i s t r ibu t ions and d i s t r ibu t ions of o rde r k

(see Sibuya et al. (1964), Pat i l and Bl ld ikar (1967), Ph i l ippou (1983, 1984), Ph i l ippou et al. (1983), Aki et al. (1984), and Ak i (1985)). We shall no t men t ion them all specifically, however, for space economy. Also, in o rder to avoid unnecessary repet i t ions , we ment ion here that in this pape r x~x . . . . . Xmk are non -nega twe integers as specified. Moreover , whenever sums and p roduc t s are t aken over t and j , ranging, respect ively, f rom 1 to m and f rom 1 to k, we shall ommi t these l imits for no ta t iona l s implici ty .

2. Multivariate distributions of order k

In this section, we ob ta in three mul t ivar ia te d i s t r ibu t ions of o rde r k by ex tend ing respect ive results of Phi l ippou (1987) to the mul t iva r ia te case. F i rs t we cons ider the fol lowing

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Propos i t ion 2.1. A n urn con tams balls bearing the letters F n . . . . . Fmk and S ( = Soo) with respective

propornons q n . . . . . qmk and p (0 < qu < 1 fo~ 1 <~ t <~ m and 1 <~j <~ k, ql l q: " " " +qmk < 1 and q n + " " " +q , ,k + P = 1). Balls are drawn f r o m the ur with replacement unttl r balls (r >1 1) bearmg the letter S

appear. Le t X, (1 <~ t <~ m ) be a random vartable denonng the sum o f the second radices o f the letters on the balls drawn whose f i rs t mdtce ts t. Then

P ( X ~ = x , . . . . . X , . = x , . ) = p r E ~ , ] J X l j ~ X t

1 = 1 , , m

( ) Tj 1 0 0 1 270.96 637.20 Tm 1 1 1 rg /F0 8.88 Tf 0 Tc 0 Tw 93 Tz (x ) Tj 6 0 TD 1 1 1 rg 1.24 Tc 0 Tw (n+ ) Tj 21.12 0 TD 1 1 1 rg 1.24 Tc 0 Tw (... ) Tj 16.56 0 TD 1 1 1 rg 1.24 Tc 0 Tw (+xmk+r-1 ) Tj 1 0 0 1 281.04 622.08 Tm 1 1 1 rg /F3 5.52 Tf 0.77 Tc 0 Tw 89 Tz (Xll,...~Xmk ) Tj 52.56 0 TD 1 1 1 rg 0 Tc 0 Tw (~ ) Tj 6.72 0 TD 1 1 1 rg 0.77 Tc 0 Tw (r-- ) Tj 16.08 0 TD 1 1 1 rg /F1 8.40 Tf 0 Tc 0 Tw 93 Tz (1 ) Tj ET BT 3 Tr 1 0 0 1 373.44 628.32 Tm 1 1 1 rg /F6 34.80 Tf 0 Tc 0 Tw 100 Tz () qqXl~ . . . x~k a m k

x 1 = 0 , 1 . . . . . . . . . X m = O , 1 . . . . .

Proof . F o r any f ixed non-negat ive integers x 1 . . . . . x , , , a typmal e lement of the event (X~ = x I . . . . . X m = x , , ) is an a r rangement a l a 2 • • • ax, ' + m - •, - +x k+, 1 S ° f t h e l e t t e r s F n , - " F,, k and S, s u c h t h a t r 1 o f the a ' s a r e S , x u of the a ' s a r e F , u ( l ~ l ~ m a n d l ~ < y ~ k ) , a n d

1 L , j x u = x , , t = . . . . . m. t z . t ) J

Fix x n . . . . . Xmk ( r is fixed). Then the n u m b e r of the above a r rangements is

Volume 7, Number 3 STATISTICS & PROBABILITY LETTERS December 1988

Proposition 2.1, specialized to the case r = 1, provides a genesis scheme for MGk(qH . . . . . qmk)" Furthermore, if qu = p / - 1 Q , (Q, = 1 - P,, 1 ~ t ~< m and 1 ~<j ~< k) so that p = 1 - Z,(1 - p k) -- p , we

observe that MNBk(r; qal . . . . . qmk) reduces to the following multivariate negative binomial distribution of

order k:

r ( r + Y.,~jx,j ) e ( X l = x l . . . . . X m = x ' ) = P r ~" r(r)II,17~x,~!

~ j J X i j ~ x~

x , = 0 , 1 . . . . . t<~t<~m,

n,p:,(e,l P, ] ' (2.3)

which is the multivariate analogue of the (shifted) negative binomial distribution of order k of Philippou et al. (1983) (see, also, Aki et al. (1984)). We call it multivariate negative binomial distribution of order k, type I, with parameters r, Q1 . . . . . Qm, and denote it by MNBk,I(r; Q1 . . . . . Q,,). For r = 1, (2.3) reduces to a multivariate distribution of order k, which we call (shifted) multivariate geometric distribution of order k, type I, with parameters Q1 . . . . . Qm and denote by MGk,I(Q 1 . . . . . Qm)- It is the multivariate analogue of the (shifted) geometric distribution of order k. We write

MNBk,,(1; Q, . . . . . Q,.) = MGk. , (Q , . . . . . Qm). (2.4)

If q , j = Q , / k ( l ~ < t ~ < m and z~<j~<k) so that p = l - Q ~ . . . . . Q , , = P , we note that MNBk(r ; qn . . . . . qmk) reduces to the following multivariate negative binomial distribution of order k:

F( r + w~,,~jx,, ) P ( X I = x 1 . . . . . X m = X m ) = P r Z

"~jJXtj = X 1

xl = 0, 1 , . . . , l<~t<<.m,

which is the multivariate analogue of the compound Poisson (or negauve binomial) distribution of order k of Philippou (1983). We call it multivariate negative binomial distribution of order k, type II, with parameters r, Q1 . . . . . Qm, and denote it by MNBk,n(r; Q1 . . . . . Qm)- For r = 1, (2.5) reduces to a multivariate distribution of order k, which we call multivariate geometric distribution of order k, type II, and denote by MGk,n(Qa . . . . . Qm). We write

MNBk,I,(q; Q1 . . . . . Qm) = MGk,n(Q1 . . . . . Qm)- (2.6)

It is obvious from the above deliberations that Proposition 2.1 provides a genesis scheme for each one of the type I and type II multivariate negative binomial distributions of order k (and hence for

MGk.I(Q1 . . . . . Qm) and MGk,n(Q1 . . . . . Qk), as well). We proceed now to derive a multivariate Poisson distribution of order k as a limiting case of the

multivariate negative binomial distribution of the same order.

Theorem 2.3. Let X , , r > 0, be m X 1 random vectors dzstrtbuted as MNBk(r; qll . . . . . q,,k ), and assume that q,j ~ 0 and rqu ~ )Lj (0 < )~,j < 0o for 1 <~ t <~ m and 1 <~ j <~ k ) as r ~ o¢. Then for x, = 0, 1 . . . . . 1 ~< I ~< m, we have

n,ny, y P ( X, = x , . . . . . X,,,= x,,,) ~ ~_, e x p ( - ~ , ~ j k , s ) FI,Hjxu~ "

~jJX t j ~ X t

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Proof. For x, = 0, 1 . . . . . 1 ~< i ~< m, we have

r'( r + ~,,~tx,j ) P(X1 = X 1 . . . . . Xm=Xm)__pr E

Zjx,, =x, -~'( r ) 1 7 , H j x u ! I-l'I-I'q'~f'

( rZ, Zjqu)~ F(r+.~,Zjx , j ) = 1 - Y~.

r ,~#x,~=x, F ( r ) r~'~j~',

n,n,x ,y ~ e x p ( - Z , Z , ) L , ) E n , n , x , , ! '

~ j J X t j = X t

which establishes'the proposmon.

n ,n j ( r%) x,' n ,17 jx ,j !

Definition 2.4. A random vector X = ( X 1 . . . . . X m) is said to have the multivariate Poisson distribution of order k with parameters X]I . . . . . X,,k (0~<X, j<o0 for l~<t~<m and l ~ < j ~ k ) , to be denoted by M P k ( X n . . . . . X m k ) , if

e ( x 1 = X 1 . . . . . X m = x..) - - E exp(-rlr , X,,) n,nyy r_#x,,=x, l"l,l'-ljx,j! ' x, = 0, 1 . . . . . 1 ~< t ~< m.

For k = 1, this distribution reduces to the usual multivariate or multiple Poisson distribution (see, e.g. Johnson and Kotz (1969, p. 197), and Feller (1968, p. 172)). For m = 1, it reduces to the mult iparameter Polsson distribution of order k of Philippou (1987). The latter was called by Aki (1985) extended Poisson distribution of order k, since it extends the Poisson distribution of order k, and therefore, MPk(X]I . . . . . hmk ) may also be called multivariate extended Poisson distribution of order k.

The following corollaries to Theorem 2.3, regarding the multivariate negative binomial distributions of order k, type I and type II, are obvious.

Corollary 2.5. Let X,, r > 0, be m x 1 random vectors chstrtbuted as MNBka(r ; Q1 . . . . . Qm), and assume that Q, ~ O and rQ, ~ X , ( 0 < ) % < oc for l <~ l <~ m ) as r ~ oo. Then, for x, = O, 1 . . . . . 1 <~ t <~ m, we have

I ! ~ t 'X tJ

P ( X, = x a . . . . . X m = X,,) ~ E e x p ( - k Z , X,) F1,H, xu ! . ~, jJXi j ~ X t

Corollary 2.6. Let X r, r > 0, be m x 1 random vectors dtstrtbuted as MNBk,u(r; Q1 . . . . . Q,,), and assume that Q , ~ 0 and r Q , ~ k X , ( 0 < h , < ~ for 1 <~t<~m) as r ~ oo. Then, for x , = 0 , 1 . . . . . 1 <~t<~m, we have

P ( X~ = x, . . . . . X , , = xm) ~ E e x p ( - k E , X,) l-I'h'EJx- Z#x,~ = x, l-l,l--Ijx,j ! "

We note that Corollary 2.5 generalizes Theorem 3.2 of Philippou et al. (1983) to the multivariate case. We call the limit distribution m the above two corollaries multivariate Po~sson distribution of oder k, type I, with parameters X] . . . . . X,, (0 ~< X, ~< oo for 1 ~ t ~ m), and denote it by MPk,I(Xa . . . . . Am)- If X u = X, (1 ~< t ~< m) for 1 ~ j ~< k, then

MPk(XI, . . . . . Xm, ) = MPk,I(X , . . . . . X,.).

The multivariate negative binomial distribution of order k may also give, in the limit, the multivariate logarithnnc series distribution of the same order.

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Theorem 2.7. Let X r, r > O, be m x 1 random vectors dtstrtbuted as M N B k ( r ; q l l . . . . . qmk), and assume that r --" O. Then, for x, = O, 1 . . . . . 1 <~ i <~ m, and ~.,x, > O, we have

P(X,=x , .. . . . Xm=XmIZ, X,>O)--" E (Z, Zjx,s--1)! Z~,~,,=~, I1,17jx,,! 1717jq,~"

where a --- - (log p ) - 1.

Proof . F o r x, = 0, 1 . . . . . 1 ~< ~ ~< m, and Z , x , > 0, we have

P( X, = x, .. . . . X~= x~lZ, X,>o)

P ( X , = X 1 . . . . . X m = x m , z , g t > 0 )

1 -P(X~ = 0 . . . . . X , . = O)

rp"

1 -p" Y] ~ j J X t j ~ X I

p___~ r ( r + Z,Zjx,~) = 1 _pr E r Ir ) ILU~%! 1LILq'~"

~ j J X t j = X t

F ( r + Z, Z j x , j ) n , Usq,~,,

rr ( r ) n,n~x,,!

Fl, IIjq,Xj ~

---, - ( l o g p ) - ' E ( 2 ~ , L x u - 1)! I l l l s x , , ! ' ~ ] J x t j = X:

which establ ishes the theorem.

Defini t ion 2.8. A r a n d o m vector X = ( X 1 . . . . . X,.) is said to have the mul t iva r i a te logar i thmic series d i s t r ibu t ion of o rde r k wi th pa rame te r s q n . . . . . q,,k ( 0 < q u < 1 for l ~ < t ~ < m and l ~ j ~ < k , a n d

qll -4- - . . q-qmk < 1), to be deno ted by M L S k ( q n . . . . . qmk), if

P(Xl=Xl .. . . . Xm=Xm) =tl E ( Z ' ~ x u - 1)! zjx,j=x, II, Iljxu! II17jq,~',,

X , = 0 , 1 . . . . . l < t < ~ m , and E , x , > 0 ,

where a = - ( l o g p ) - i and p = 1 - q n . . . . . qmk"

F o r k = 1, this d i s t r ibu t ion reduces to the usual mul t ivar ia te logar i thmic series d i s t r ibu t ion (see, e.g. Johnson and Ko tz (1969, p. 302)). F o r m = 1, it reduces to the m u l t i p a r a m e t e r logar i thmic series

d i s t r ibu t ion of o rde r k of Ph i l ippou (1987). Fur the rmore , if qu = p / - 1 Q (Q, = 1 - P,, 1 ~ i <~ m and 1 <~j <~ k ) so that p = 1 - Z , ( q - p k) ==_ p, we

observe that M L S k ( q n . . . . . qmk) reduces to the fo l lowing mul t ivar ia te logar i thmic series d i s t r ibu t ion of o rder k:

, ~ j J X t j = X t

x , = O , 1 . . . . . l < t ~ < m , and ] 2 , x , > O ,

which is the mul t ivar ia te ana logue of the logar i thmic series d i s t r ibu t ion of o rde r k of Ak i et al. (1984). W e call it mul t ivar ia te logar i thmic series d i s t r ibu t ion of o rder k, type I, with pa r a me te r s Q1 . . . . . Q, . , a n d

denote it b y MLSk,I(Q1 . . . . . Q, . ) .

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If q,j = Q , / k (1 ~< t ~< m and 1 ~<j ~< k) so that p = 1 - Q 1 . . . . . Q,, = P, we note that MLSk(qll . . . . . q,,k) reduces to the following multivariate logarithmic series distribution of order k

P ( X 1 = x , , . . . , Xm= x,,,) = - ( l o g P)-' P( X1 = x,,..., Xm= x,,,) = -(log P)-' ~jJXIj ~ Xt

x,---O, 1 , . . . , l ~ < ~ < m , and ~7,x,>O,

(2~,2~jx,j - 1)! ) e~,,,

(2.8)

which is different from the multivariate logarithmic series distribution of order k, type I. We call it multivariate logarithmic series distribution of order k, type II, with parameters Q~ . . . . . Qm, and denote it by MLSk,H(Q1 . . . . . Qm).

The following corollaries to Theorem 2.7, regarding the multivariate negative binomial and logarithmic series distributions of order k, type I and type II, are obvious.

Corollary 2.9. Let X,, r > 0, be m × 1 random vectors dtstributed as MNBk,I(r; Q1 . . . . . Q,,), and assume

that r ~ O. Then, for x, = O, 1 . . . . . 1 <~ l <~ m, and ,~,x, > 0, we have

l . ( X , = x , . . . . . =x lZ, X, > (Z,~jx,j - 1)! ( Q,)~jx,j

n , njx , j ! 1LP/' T, '

where a = - ( log p ) - i (and P = 1 - ,~ , (1 _pk)).

Corollary 2.10. Let X,, r > 0, be m × 1 random vectors distributed as MNBk,ii(r, Qa,---, Qm), and assume that r --+ O. Then, for x, = O, 1 . . . . . 1 <~ i <~ m, and ~ , x , > O, we have

(,X,,Xjx,j- 1), Fl,17jx,j! ( _ ~ ) z j x , j P(X ,=x , ... . . Xm=x.lZ, x , > o ) - . . E 17, ~3J~I)~XI

where a = - ( log p ) - I (and P = 1 - Q1 . . . . . Qm).

It may be noted that Corollary 2.9 generalizes Proposition 3.2 of Aki et al. (1984) to the multivariate case.

3. Characteristics, genesis schemes and interrelationships

In this section, we obtain the probability generating functions, means, variances and covariances of the multivariate distributions of order k, treated in Section 2, and derive some further genesis schemes of them and interrelationships among them, by extending respective results of Philippou (1987) to the multivariate case.

We present first a genesis scheme for the multivariate negative binomial distribution of order k and a representation for each one of them, which will be useful in the sequel.

Theorem 3.1. Let X and Y be a random vector and a random variable, respectwely, such that X I Y = Y zs

distmbuted as MPk(y)~11 . . . . . Y~, , k ) and f y ( y ) = ary r-1 e - ~ Y / F ( r ) (y , a and r are posmve reals), and set

qtJ = ~ t J / ( Ol "~- A l l "~ " " " -}-~kmk) ( 1 ~ 1 ~ m and 1 <~j <~ k ) and p = 1 - qll . . . . . q,,k. Then X is distrib-

uted as MNBk(r; qll . . . . . q,,k)-

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Proof. For x, = 0, 1 . . . . . 1 ~< / ~< m, we have

P ( X , = x , . . . . . X~ = x~)

/ 7 ,n j (yX, , ) ~'' • a" y r - ~ e - " V d y

r ( r )

e x p [ - y ( a + -Y,Zj~u) ] d y

= f0 ~ E exp(-Y'X,'XjXu) i i , i l j x u ! ~ j J X t ) = X t

O0 FI,IIJX~'7 f y r - , + z , Z , x , , = +'r Z rCr)n,n,x, ,! . ~ j j x u ~ x t "tO

rl rl h~,/ r ( r + Z,Lx,~ )

-- rCr)n,n,x, ,! +

= + z , L x , , Z + z , L x , j z,j~,,= ;,,

XIj

D.

To Theorem 3.1, we have the following corollary whach extends to the multivanate case Proposition 3.1 of Philippou (1983).

Corollary 3.2. Let X and Y be a random vector and a random variable, respectively, such that X I Y = Y is dtstnbuted as MPk,I(y~] . . . . . y ~ , , ) and f r ( Y ) = arY r-1 e - ~ Y / F ( r ) (Y, a, r pos i twe reals), and set Q, = k X , / ( a + k ~ , X , ) (1 <~ t <~ m ) and P = 1 - 2~,Q,. Then X is dtstrtbuted as MNBk,n(r; Q1 . . . . . Q,,).

We also note the following representations which are direct consequences of Definitions 2.4, 2.2 and 2.8, respectively.

Proposition 3.3. Let X u, l <~ l <~ m and l < j <~ k, be rvs and set X , = Z j j X , j ( l<~l<~m) and X =

( X 1 . . . . . X, . ) . Then (a) X is dtstributed as MPk(X n . . . . . Xmk ) i f and only If X u, 1 <~ l <~ m and 1 <~j <~ k, are distributed

mdependently as P ( Au)" (b) X is distributed as MNBk(qn . . . . . qmk) i f and only i f X n . . . . . Xmk are lomt l y dtstnbuted as

multwariate negative bmomial wtth parameters r, q n , . . . , q,.k. (¢) X ts distributed as MLSk(qn . . . . . q,~k) t f and only i f X n . . . . . Xmk are joint ly &stributed as multi-

oartate logarithmic series with parameters qll . . . . , q,,k-

Proposition 3.4. Let X,, 1 ~<l~<m, be rvs and set X = ( X 1 . . . . . Xm). Then X is distributed as MPk(), n . . . . . X, ,g) t f and only t f X,, 1 <~ i ~ m, are distributed mdependently as Pk(An . . . . . h,k).

The probability generating functions (and hence the means, variances and covariances) of the multi- variate Poisson, negative binomial and logarithmic series distributions of order k may be obtained by means of the transformations x u = n u and x , = n, + X j ( j - 1)n u (1 ~< / ~< m and 1 ~<j ~< k), and this procedure will be actually used in Proposition 3.7 below. Other procedures, however, are much simpler for the first two distributions.

Upon using Proposition 3.3(a) or 3.4, we readily get

Proposition 3.5. Let X'= (X 1 . . . . . X, , ) be a random vector dtstrtbuted as M P k ( ~ l l . . . . . ~kmk ). Then

(a) g x ( t ) = e x p [ Z , Z ~ X u ( t / - 1)], I t, [ ~< 1 (1 ~< t ~< m); (b) E(X,) = Z~jA,~ and Vat(X,) = ~jJ2h, j , 1 ~< i 6 m; (c) Cov(X,, X s ) = 0 , l < ~ l ~ s < ~ m .

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U p o n using simple expecta t ion properties, Theorem 3.1 and Proposi t ion 3.5, we get par ts (a) and (b) of the following.

Proposi t ion 3.6. Let X = ( X 1 . . . . . X , . ) be a random vector dis tr ibuted as MNB~(r ; qll . . . . . qmk). Then (a) g x ( t ) = p r ( 1 - Z , Z l q , s t / ) -~, It, [ ~< 1 (1 ~< ~ ~< m); (b) E ( X , ) = ( r / p ) ~ , j j q , / and Var(X,) = ( r / p ) [ ~ , s j Z q , j + (1/p)( ,T , i jq , s )2] , 1 <~ t <~ m; (c) Cov(X, , X~) = ( r / p ) (~ , j jq , j ) ( ,Xs jq~j ) , 1 <~ t --/: s <~ m.

We get pa r t (c) by s t ra ightforward different iat ion of gx(t). We proceed now to establish the following proposi t ion, as announced earher.

Proposition 3.7. Let X = ( X 1 . . . . . Xm) be a random vector dis tr ibuted as MLSk(q l 1 . . . . . qmk)" Then

(a) g x ( t ) = - a l o g ( 1 - Z , ~ j q , s t / ) , It, [ ~< 1 (1 ~< t ~< m); (b) E ( X , ) = (a /p)~ jq , j and Var(X,) = ( a / p ) t ~ j j 2 q , j + ((1 - a ) / p ) ( ~ j q , j)2], 1 <~ z <~ m;

(c) Cov(X, , Xs) = - ( a ( a - 1 ) / p 2 ) ( Y . s j q , s ) ( Y . j q ~ j ) , 1 <~ t ~ s <~ m.

Proof . Since (b) and (c) follow f rom (a) by s t ra ightforward differentiat ion of g x ( t ) , it suffices to show (a). T o this end, let I t , 1 < 1 , x , = 0 , 1 . . . . . l ~ < ~ < m , and 2~ ,x ,>O. U p o n using Defini t ion 2.8 and the t ransformat ions x,j = n,j and x , = n, + ~ j ( j - 1)n,j (1 ~< l ~< m, 1 ~</~< k) , we get

(Z 'Y ' In ' j -1) ! . I I , FIj(q,//)~'J/ " ( Z t n , > O , 1=1 . . . . . m) g x ( t ) = a E~° " '" ~'z_, E FI, H j n , j ! nl = 0 r im=0 .Xjn,j=n,

( ,~ , ,Xjn , j - 1)! . H , F l j ( q, j t / ) , , j = ~ ~z., H , FIjn,s[

n = l Z,Zyn,3=n

= a ~ (Y"2~Jq 's t / )n

n n = l

(by the mul t inomial theorem)

= - a log(1 - ~.,~sq, j t / ) ,

which was to be shown.

Proposi t ion 3.6(a) implies the following.

Theo rem 3.8. Le t Xs , 1 <~ s <~ n, be mdependent m × 1 random vectors dts tr tbuted as MNBk(rs ; qll . . . . . q,,k), and set X = X 1 + • • • + X n and r = r 1 + • • • + r n. Then X ts dis tr ibuted as M N B k ( r ; ql l . . . . . q , , k ) .

I f r s = 1 (1 ~< s ~< n) and n = r, we get the follow/lag.

Corollary 3.9. L e t X s, 1 <~ s <~ r, be independent m X 1 random vectors dts tr lbuted as M G k ( q l 1 . . . . . qmk), and set X = X 1 + • • • +3(, . Then X ts dis tr ibuted as MNBk( r ; qll . . . . . qmk)"

The specialization of Corol lary 3.9 to MGk,I(Q1 . . . . . Q, , ) is the mul t ivar ia te analogue of the shifted version of Theo rem 3.1 of Phi l ippou et al. (1983).

I t is well known that the negative b inomial dis t r ibut ion results if the Poisson dis t r ibut ion with mean - r log p is generalized by the logar i thmic series distribution. We shall now show that this scheme carries over to the mult ivar ia te negative b inomial and logar i thmic series distr ibutions of order k.

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Theorem 3.10. Let X+, s >/1, be mdependent m × 1 random vectors distributed as MLSk(ql a . . . . . q,,k) independently of a random variable N which ts dtstrtbuted as P ( - r log p), and set S N = X 1 + • • • + X N.

Then SN IS dtstributed as MNBk(r; qxt . . . . . qmk).

Proof. For It, I ~< 1 (1 ~< l ~< m), we have

g s N ( t ) = g N [ g x , ( t ) ] = e x p { - - r log p [ g x , ( t ) - - 1 ] }

= e x p { - - r log p [ - - a l o g ( 1 - Z , Z j q u t ] ) - 1]} (by Proposition 3.7(a))

= e x p { - r [ l o g ( 1 - ~,~,jq, j t ] ) - log , ] }

= p ' ( 1 - - Z , • q , j t ] ) - ' ,

which establishes the theorem by means of Proposition 3.6(a). It was mentioned in Section 2 that MPk(X H . . . . . hmk ) might have been called multivariate extended

Poisson distribution of order k. Since MNBk(r; qH . . . . . qmk) is a gamma mixture of MPk(yXll . . . . . yh, ,k) , and MLSk(qll . . . . . q,,k) is a limit of the first, the latter two distributions might have been called multivariate extended negative binomial and logarithmic series distributions of order k.

We shall give now the multivariate analogues of the (suitably shifted) extended negative binomial and logarithmic series distributions of order k of Aki (1985).

Let X = ( X 1 . . . . . Xm) be a random vector distributed as MNB~(r; qal . . . . . q,,k)- We consider the transformation

qa = Q,1 and q,j = Pa "'" P , j - 1 Q , j ( Q , j = 1 - P, j ) , 1 <~t <~ m, 2 <~j <~ k , (3.1)

and set Q = ~,(17jP, j ) - (m - 1). Then, for x, = 0, 1 . . . . . 1 ~< t ~< m, we get

r ( r + ~ , L x , j ) H , H j ( 1 - P, )X"H,H~-IP,5:=< . . . . . , P ( X 1 = x t . . . . . X m = x m ) = Q r E r ( r ) H l - I j x , , t J =

~ j J X t j ~ X t

(3.2)

which reduces to the (suitably shifted) extended negative binomial distribution of order k of Aki (1985) for m = 1 and any positive integer r. We call (3.2) multivariate (suitably shifted) extended negative binomial distribution of order k with parameters r, Pl l . . . . . Pink, and denote it by M E N B k ( r ; Pit . . . . . Pink)- For r = 1, (3.2) reduces to a multivariate (suitably shifted) extended distribution of order k, which we call multivariate (suitably shifted) extended geometric distribution of order k with parameters Pla . . . . . P,,k and denote by MEGk(Plt . . . . . Pink)" We write

MENBk(1; Pit . . . . . Pmk) = MEGk(P11 . . . . . P,,k)" (3.3)

Let now X = (X~ . . . . . Arm) be a random vector distributed as MLSk(q H . . . . . qmk)- We consider again the transformation (3.1) and set Q as above. Then, for x, = 0, 1 . . . . . 1 ~< t ~< m, and Z,x, > 0, we get

P( = x, . . . . . Xm= Xm)

= - ( l o g Q ) - I y , (Y . ,~ jx , j - 1)! .,++,.,.,,,,-x, n,n+,x,.,t n,l-i ,(1 . . . . .

which reduces to the extended logarithmic series distribution of order k of Aki (1985) for m = 1. We call (3.4) multivariate extended logarithmic series distribution of order k with parameters Paa . . . . . P,,k, and denote it by MELSk(Pla . . . . . P,,k)-

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Since the multivariate Poisson distribution of order k is a multivariate analogue of the extended Poisson distribution of order k, as mentioned in Section 2, and the multivariate negative binomial and logarithmic series distributions of order k are multivariate analogues of the extended negative binomial and logarithmic series distributions of the same order, because of (3.1), (3.2) and (3.4), it is obvious that the present work generalizes to the multivariate case several results of Aki (1985) and Hirano and Aki (1987). For example, Theorem 3.10 generalizes Proposition 2a of the latter authors to the multivariate case.

Further properties of the multivariate distributions of order k will be reported soon, including the multivariate modified logarithmic series distribution of the same order.

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