316 Some Elementary Statistical Inferencesthe random interval, can we be led to an additional distribution-free method of
statistical inference?
LetXbe a random variable with distribution functionF(x) of the continuous
type. LetZ=F(X). Then, as shown in Exercise 4.8.1,Zhas a uniform(0,1)
distribution. That is,Z=F(X)hasthepdf
h(z)={
10 <z< 1
0elsewhere.Then, if 0<p<1, we have
P[F(X)≤p]=∫p0dz=p.NowF(x)=P(X≤x). SinceP(X=x)=0,thenF(x) is the fractional part of
the probability for the distribution ofXthat is between−∞andx.IfF(x)≤p,
then no more than 100p% of the probability for the distribution ofXis between
−∞andx. But recallP[F(X)≤p]=p. That is, the probability that the random
variableZ=F(X)islessthanorequaltopis precisely the probability that the
random interval (−∞,X) contains no more than 100p% of the probability for the
distribution. For example, ifp=0.70, the probability that the random interval
(−∞,X) contains no more than 70% of the probability for the distribution is 0.70;
and the probability that the random interval (−∞,X) contains more than 70% of
the probability for the distribution is 1− 0 .70 = 0.30.
We now consider certain functions of the order statistics. LetX 1 ,X 2 ,...,Xn
denote a random sample of sizenfrom a distribution that has a positive and con-
tinuous pdff(x) if and only ifa<x<b,andletF(x) denote the associated distri-
bution function. Consider the random variablesF(X 1 ),F(X 2 ),...,F(Xn). These
random variables are independent and each, in accordance with Exercise 4.8.1, has
a uniform distribution on the interval (0,1). Thus,F(X 1 ),F(X 2 ),...,F(Xn)isa
random sample of sizenfrom a uniform distribution on the interval (0,1). Consider
the order statistics of this random sampleF(X 1 ),F(X 2 ),...,F(Xn). LetZ 1 be the
smallest of theseF(Xi),Z 2 the nextF(Xi) in order of magnitude,... ,andZn
the largest ofF(Xi). IfY 1 ,Y 2 ,...,Ynare the order statistics of the initial random
sampleX 1 ,X 2 ,...,Xn,thefactthatF(x) is a nondecreasing (here, strictly increas-
ing) function ofximplies thatZ 1 =F(Y 1 ),Z 2 =F(Y 2 ),...,Zn=F(Yn). Hence, it
follows from (4.4.1) that the joint pdf ofZ 1 ,Z 2 ,...,Znis given by
h(z 1 ,z 2 ,...,zn)={
n!0<z 1 <z 2 <···<zn< 1
0elsewhere.
(4.10.1)This proves a special case of the following theorem.
Theorem 4.10.1.LetY 1 ,Y 2 ,...,Yndenote the order statistics of a random sample
of sizenfrom a distribution of the continuous type that has pdff(x)and cdfF(x).
The joint pdf of the random variablesZi =F(Yi),i=1, 2 ,...,n, is given by
expression (4.10.1).