Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

452 Sufficiency

hence,K 1 (x)=x. The first term is easily seen to be a linear combination of the
productsxixj,i, j=1, 2 ,...,k, which are the entries of the matrixxx′. Hence we
can takeK 2 (x)=xx′.Now,letX 1 ,...,Xnbe a random sample onX. Based on
(7.7.7) then, a set of sufficient and complete statistics is given by

Y 1 =

∑n

i=1

XiandY 2 =

∑n

i=1

XiX′i. (7.7.8)

Note thatY 1 is a vector ofkstatistics and thatY 2 is ak×ksymmetric matrix.
Because the matrix is symmetric, we can eliminate the bottom-half [elements (i, j)
withi>j] of the matrix, which results in{k+[k(k+1)]}complete sufficient
statistics, i.e., as many complete sufficient statistics as there are parameters.
Based on marginal distributions, it is easy to show thatXj=n−^1

∑n
i=1Xijis
the MVUE ofμj and that (n−1)−^1

∑n

i=1(Xij−Xj)

(^2) is the MVUE ofσ 2
j.The
MVUEs of the covariance parameters are obtained in Exercise 7.7.9.
For our last example, we consider a case where the set of parameters is the cdf.
Example 7.7.5.LetX 1 ,X 2 ,...,Xnbe a random sample having the common con-
tinuous cdfF(x). LetY 1 <Y 2 <···<Yndenote the corresponding order statistics.
Note that givenY 1 =y 1 ,Y 2 =y 2 ,...,Yn=yn, the conditional distribution of
X 1 ,X 2 ,...,Xnis discrete with probabilityn^1! on each of then! permutations of
the vector (y 1 ,y 2 ,...,yn), [becauseF(x) is continuous, we can assume that each
of the valuesy 1 ,y 2 ,...,ynis distinct]. That is, the conditional distribution does
not depend onF(x). Hence, by the definition of sufficiency, the order statistics are
sufficient forF(x). Furthermore, while the proof is beyond the scope of this book,
it can be shown that the order statistics are also complete; see page 72 of Lehmann
and Casella (1998).
LetT=T(x 1 ,x 2 ,...,xn)beanystatisticthatissymmetric in its arguments;
i.e.,T(x 1 ,x 2 ,...,xn)=T(xi 1 ,xi 2 ,...,xin) for any permutation (xi 1 ,xi 2 ,...,xin)
of (x 1 ,x 2 ,...,xn). ThenTis a function of the order statistics. This is useful in
determining MVUEs for this situation; see Exercises 7.7.12 and 7.7.13.
EXERCISES
7.7.1.LetY 1 <Y 2 <Y 3 be the order statistics of a random sample of size 3 from
the distribution with pdf
f(x;θ 1 ,θ 2 )=
{
1
θ 2 exp
(
−x−θ 2 θ^1
)
θ 1 <x<∞, −∞<θ 1 <∞, 0 <θ 2 <∞
0elsewhere.
Find the joint pdf ofZ 1 =Y 1 ,Z 2 =Y 2 ,andZ 3 =Y 1 +Y 2 +Y 3. The corresponding
transformation maps the space{(y 1 ,y 2 ,y 3 ):θ 1 <y 1 <y 2 <y 3 <∞}onto the
space
{(z 1 ,z 2 ,z 3 ):θ 1 <z 1 <z 2 <(z 3 −z 1 )/ 2 <∞}.
Show thatZ 1 andZ 3 are joint sufficient statistics forθ 1 andθ 2.