Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.8. Minimal Sufficiency and Ancillary Statistics 457

(c)Supposef(w) is the logistic pdf. As discussed in Example 6.1.2, the mle ofθ

exists and it is easy to compute. As shown on page 38 of Lehmann and Casella

(1998), though, the order statistics are minimal sufficient for this situation.

That is, no reduction is possible.

(d)Supposef(w) is the Laplace pdf. It was shown in Example 6.1.1 that the

median,Q 2 is the mle ofθ, but it is not a sufficient statistic. Further, similar

to the logistic pdf, it can be shown that the order statistics are minimal

sufficient for this situation.

In general, the situation described in parts (c) and (d), where the mle is obtained

rather easily while the set of minimal sufficient statistics is the set of order statistics

and no reduction is possible, is the norm for location models.

There is also a relationship between a minimal sufficient statistic and complete-

ness that is explained more fully in Lehmann and Scheff ́e (1950). Let us say simply

and without explanation that for the cases in this book, complete sufficient statistics

are minimal sufficient statistics. The converse is not true, however, by noting that

in Example 7.8.1, we have

E

[

Yn−Y 1

2

−

n− 1

n+1

]

=0, for allθ.

That is, there is a nonzero function of those minimal sufficient statistics,Y 1 andYn,
whose expectation is zero for allθ.
There are other statistics that almost seem opposites of sufficient statistics.
That is, while sufficient statistics contain all the information about the parameters,
these other statistics, calledancillary statistics, have distributions free of the
parameters and seemingly contain no information about those parameters. As an
illustration, we know that the varianceS^2 of a random sample fromN(θ,1) has
a distribution that does not depend uponθand hence is an ancillary statistic.
Another example is the ratioZ=X 1 /(X 1 +X 2 ), whereX 1 ,X 2 is a random sample
from a gamma distribution with known parameterα>0 and unknown parameter
β=θ, becauseZhas a beta distribution that is free ofθ. There are many examples
of ancillary statistics, and we provide some rules that make them rather easy to find
with certain models, which we present in the next three examples.

Example 7.8.4(Location-Invariant Statistics). In Example 7.8.3, we introduced
the location model. Recall that a random sampleX 1 ,X 2 ,...,Xnfollows this model
if
Xi=θ+Wi,i=1,...,n, (7.8.3)

where−∞<θ<∞is a parameter andW 1 ,W 2 ,...,Wnare iid random variables
with the pdff(w), which does not depend onθ. Then the common pdf ofXiis
f(x−θ).
LetZ=u(X 1 ,X 2 ,...,Xn)beastatisticsuchthat

u(x 1 +d, x 2 +d,...,xn+d)=u(x 1 ,x 2 ,...,xn),