Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.10.∗Tolerance Limits for Distributions 317

Because the distribution function ofZ=F(X)isgivenbyz, 0 <z<1, it
follows from (4.4.2) that the marginal pdf ofZk=F(Yk) is the following beta pdf:

hk(zk)=

{ n!

(k−1)!(n−k)!z

k− 1

k (1−zk)

n−k 0 <zk< 1

0elsewhere.

(4.10.2)

Moreover, from (4.4.3), the joint pdf ofZi=F(Yi)andZj=F(Yj)is,withi<j,
given by

h(zi,zj)=

{

n!zii−^1 (zj−zi)j−i−^1 (1−zj)n−j

(i−1)!(j−i−1)!(n−j)!^0 <zi<zj<^1

0elsewhere.

(4.10.3)

Consider the differenceZj−Zi=F(Yj)−F(Yi),i<j.NowF(yj)=P(X≤yj)
andF(yi)=P(X≤yi). SinceP(X=yi)=P(X=yj) = 0, then the difference
F(yj)−F(yi) is that fractional part of the probability for the distribution ofXthat
is betweenyiandyj.Letpdenote a positive proper fraction. IfF(yj)−F(yi)≥p,
then at least 100p% of the probability for the distribution ofXis betweenyiand
yj.Letitbegiventhatγ=P[F(Yj)−F(Yi)≥p]. Then the random interval
(Yi,Yj) has probabilityγof containing at least 100p% of the probability for the
distribution ofX.Nowifyiandyjdenote, respectively, observational values ofYi
andYj,theinterval(yi,yj) either does or does not contain at least 100p%ofthe
probability for the distribution ofX. However, we refer to the interval (yi,yj)as
a 100γ%tolerance intervalfor 100p% of the probability for the distribution of
X. In like vein,yiandyjare called the 100γ%tolerance limitsfor 100p%ofthe
probability for the distribution ofX.
One way to compute the probabilityγ=P[F(Yj)−F(Yi)≥p]istouseequation
(4.10.3), which gives the joint pdf ofZi=F(Yi)andZj=F(Yj). The required
probability is then given by

γ=P(Zj−Zi≥p)=

∫ 1 −p

0

[∫ 1

p+zi

hij(zi,zj)dzj

]

dzi.

Sometimes, this is a rather tedious computation. For this reason and also for the
reason thatcoverages are important in distribution-free statistical inference, we
choose to introduce at this time the concept of coverage.
Consider the random variablesW 1 =F(Y 1 )=Z 1 ,W 2 =F(Y 2 )−F(Y 1 )=
Z 2 −Z 1 ,andW 3 =F(Y 3 )−F(Y 2 )=Z 3 −Z 2 ,...,Wn =F(Yn)−F(Yn− 1 )=
Zn−Zn− 1. The random variableW 1 is called acoverageof the random interval
{x:−∞<x<Y 1 }and the random variableWi,i=2, 3 ,...,n, is called acoverage
of the random interval{x:Yi− 1 <x<Yi}. We find that the joint pdf of then
coveragesW 1 ,W 2 ,...,Wn. First we note that the inverse functions of the associated
transformation are given by

zi=

∑i

j=1

wj,fori=1, 2 ,...,n.