Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

464 Sufficiency

The joint pdf ofY 1 andYnis

g(y 1 ,yn;θ)=n(n−1)(yn−y 1 )n−^2 / 2 n,θ− 1 <y 1 <yn<θ+1,

zero elsewhere. Accordingly, the joint pdf ofT 1 andT 2 is, since the absolute value

of the Jacobian equals 1,

h(t 1 ,t 2 ;θ)=n(n−1)tn 2 −^2 / 2 n,θ−1+

t 2

2

<t 1 <θ+1−

t 2

2

, 0 <t 2 < 2 ,

zero elsewhere. Thus the pdf ofT 2 is

h 2 (t 2 ;θ)=n(n−1)tn 2 −^2 (2−t 2 )/ 2 n, 0 <t 2 < 2 ,

zero elsewhere, which, of course, is free ofθasT 2 is an ancillary statistic. Thus,
the conditional pdf ofT 1 ,givenT 2 =t 2 ,is

h 1 | 2 (t 1 |t 2 ;θ)=

1

2 −t 2

,θ−1+

t 2

2

<t 1 <θ+1−

t 2

2

, 0 <t 2 < 2 ,

zero elsewhere. Note that this is uniform on the interval (θ−1+t 2 / 2 ,θ+1−t 2 /2);
so the conditional mean and variance ofT 1 are, respectively,

E(T 1 |t 2 )=θ and var(T 1 |t 2 )=

(2−t 2 )^2

12

.

GivenT 2 =t 2 , we know something about the conditional variance ofT 1 .Inparticu-
lar, if that observed value ofT 2 is large (close to 2), then that variance is small and
we can place more reliance on the estimatorT 1. On the other hand, a small value
oft 2 means that we have less confidence inT 1 as an estimator ofθ.Itisextremely
interesting to note that this conditional variance does not depend upon the sample
sizenbut only on the given value ofT 2 =t 2. As the sample size increases,T 2 tends
to become larger and, in those cases,T 1 has smaller conditional variance.

While Example 7.9.5 is a special one demonstrating mathematically that an
ancillary statistic can provide some help in point estimation, this does actually
happen in practice, too. For illustration, we know that if the sample size is large
enough, then

T=

X−μ

S/

√

n

has an approximate standard normal distribution. Of course, if the sample arises
from a normal distribution,XandSare independent andThas at-distribution with
n−1 degrees of freedom. Even if the sample arises from a symmetric distribution,
XandSare uncorrelated andThas an approximatet-distribution and certainly an
approximate standard normal distribution with sample sizes around 30 or 40. On
the other hand, if the sample arises from a highly skewed distribution (say to the
right), thenXandSare highly correlated and the probabilityP(− 1. 96 <T < 1 .96)
is not necessarily close to 0.95 unless the sample size is extremely large (certainly
much greater than 30). Intuitively, one can understand why this correlation exists if