Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.7. The Case of Several Parameters 451

that outcomejoccurs ispj; hence,

∑k

j=1pj=1. LetX=(X^1 ,...,Xk−^1 )

′and

p =(p 1 ,...,pk− 1 )′. The distribution ofXis multinomial and can be found in
expression (6.4.18), which can be reexpressed as

f(x,p)=exp

⎧

⎨

⎩

k∑− 1

j=1

(

log

[

pj

1 −

∑

i =kpi

])

xj+log

⎛

⎝ 1 −

∑

i =k

pi

⎞

⎠

⎫

⎬

⎭

.

Because this a regular case of the exponential family, the following statistics, re-
sulting from a random sampleX 1 ,...,Xnfrom the distribution ofX,arejointly
sufficient and complete for the parametersp=(p 1 ,...,pk− 1 )′:

Yj=

∑n

i=1

Xij, forj=1,...,k− 1.

Each random variableXijis Bernoulli with parameterpj and the variablesXij
are independent fori=1,...,n. Hence the variablesYjare binomial(n, pj)for
j=1,...,k. Thus the MVUE ofpjis the statisticn−^1 Yj.
Next, we shall find the MVUE ofpjpl,forj =l. Exercise 7.7.8 shows that the
mle ofpjplisn−^2 YjYl. Recall from Section 3.1 that the conditional distribution of
Yj,givenYl,isb[n−Yl,pj/(1−pl)]. As an initial guess at the MVUE, consider the
mle, which, as shown by Exercise 7.7.8, isn−^2 YjYl. Hence

E[n−^2 YjYl]=

1

n^2

E[E(YjYl|Yl)] =

1

n^2

E[YlE(Yj|Yl)]

=

1

n^2

E

[

Yl(n−Yl)

pj

1 −pl

]

=

1

n^2

pj

1 −pl

{E[nYl]−E[Yl^2 ]}

=

1

n^2

pj

1 −pl

{n^2 pl−npl(1−pl)−n^2 p^2 l}

=

1

n^2

pj

1 −pl

npl(n−1)(1−pl)=

(n−1)

n

pjpl.

Hence the MVUE ofpjplisn(n^1 −1)YjYl.

Example 7.7.4(Multivariate Normal).LetXhave the multivariate normal distri-
butionNk(μ,Σ), whereΣis a positive definitek×kmatrix. The pdf ofXis given
in expression (3.5.16). In this caseθis a{k+[k(k+1)/2]}-dimensional vector whose

firstkcomponents consist of the mean vectorμand whose lastk(k 2 +1)components
consist of the componentwise variancesσi^2 and the covariancesσij,forj≥i.The
density ofXcan be written as

fX(x)=exp

{

−

1

2

x′Σ−^1 x+μ′Σ−^1 x−

1

2

μ′Σ−^1 μ−

1

2

log|Σ|−

k

2

log 2π

}
,
(7.7.7)
forx∈Rk.Hence, by (7.7.5), the multivariate normal pdf is a regular case of the
exponential class of distributions. We need only identify the functionsK(x). The
second term in the exponent on the right side of (7.7.7) can be written as (μ′Σ−^1 )x;