Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.9. Sufficiency, Completeness, and Independence 465

the underlying distribution is highly skewed to the right. WhileShas a distribution
free ofμ(and hence is an ancillary), a large value ofSimplies a large value ofX,
since the underlying pdf is like the one depicted in Figure 7.9.1. Of course, a small
value ofX(say less than the mode) requires a relatively small value ofS.This
means that unlessnis extremely large, it is risky to say that

x−

1. 96 s

√

n

, x+

1. 96 s

√

n

provides an approximate 95% confidence interval with data from a very skewed
distribution. As a matter of fact, the authors have seen situations in which this
confidence coefficient is closer to 80%, rather than 95%, with sample sizes of 30 to
40.

f(x)

x

Figure 7.9.1:Graph of a right skewed distribution; see also Exercise 7.9.14.

EXERCISES

7.9.1.LetY 1 <Y 2 <Y 3 <Y 4 denote the order statistics of a random sample
of size n= 4 from a distribution having pdff(x;θ)=1/θ, 0 <x<θ, zero
elsewhere, where 0<θ<∞. Argue that the complete sufficient statisticY 4 forθ
is independent of each of the statisticsY 1 /Y 4 and (Y 1 +Y 2 )/(Y 3 +Y 4 ).
Hint: Show that the pdf is of the form (1/θ)f(x/θ), wheref(w)=1, 0 <w<1,
zero elsewhere.

7.9.2.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample from a
N(θ, σ^2 ),−∞<θ<∞, distribution. Show that the distribution ofZ=Yn−X
does not depend uponθ.ThusY=

∑n
1 Yi/n, a complete sufficient statistic forθis
independent ofZ.

7.9.3. LetX 1 ,X 2 ,...,Xnbe iid with the distributionN(θ, σ^2 ), −∞<θ<∞.
Prove that a necessary and sufficient condition that the statisticsZ=

∑n
1 aiXiand
Y=

∑n

1 Xi, a complete sufficient statistic forθ, are independent is that

∑n

1 ai=0.