Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

456 Sufficiency

From these examples we see that the minimal sufficient statistics do not need
to be unique, for any one-to-one transformation of them also provides minimal
sufficient statistics. The linkage between minimal sufficient statistics and the mle,
however, does not hold in many interesting instances. We illustrate this in the next
two examples.

Example 7.8.2.Consider the model given in Example 7.8.1. There we noted that
Y 1 =min(Xi)andYn=max(Xi) are joint sufficient statistics. Also, we have

θ− 1 <Y 1 <Yn<θ+1

or, equivalently,
Yn− 1 <θ<Y 1 +1.
Hence, to maximize the likelihood function so that it equals (^12 )n,θcan be any
value betweenYn−1andY 1 + 1. For example, many statisticians take the mle to
be the mean of these two endpoints, namely,

θˆ=Yn−1+Y^1 +1

2

=

Y 1 +Yn

2

,

which is the midrange. We recognize, however, that this mle is not unique. Some

might argue that sinceθˆis an mle ofθand since it is a function of the joint sufficient

statistics,Y 1 andYn,forθ, it is a minimal sufficient statistic. This is not the case at

all, forθˆis not even sufficient. Note that the mle must itself be a sufficient statistic

for the parameter before it can be considered the minimal sufficient statistic.

Note that we can model the situation in the last example by

Xi=θ+Wi, (7.8.1)

whereW 1 ,W 2 ,...,Wnare iid with the common uniform(− 1 ,1) pdf. Hence this is

an example of a location model. We discuss these models in general next.

Example 7.8.3.Consider a location model given by

Xi=θ+Wi, (7.8.2)

whereW 1 ,W 2 ,...,Wnare iid with the common pdff(w) and common continuous
cdfF(w). From Example 7.7.5, we know that the order statisticsY 1 <Y 2 <···<Yn
are a set of complete and sufficient statistics for this situation. Can we obtain a
smaller set of minimal sufficient statistics? Consider the following four situations:

(a)Supposef(w)istheN(0,1) pdf. Then we know thatXis both the MVUE

and mle ofθ.Also,X=n−^1

∑n

i=1Yi, i.e., a function of the order statistics.

HenceXis minimal sufficient.

(b)Supposef(w)=exp{−w},forw>0, zero elsewhere. Then the statisticY 1 is

a sufficient statistic as well as the mle, and thus is minimal sufficient.