254 Some Elementary Statistical Inferencesimportant role in statistical inference partly because some of their properties do not
depend upon the distribution from which the random sample is obtained.
LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution of thecontinu-
ous typehaving a pdff(x) that has supportS=(a, b), where−∞ ≤a<b≤∞.
LetY 1 be the smallest of theseXi,Y 2 the nextXiin order of magnitude,...,and
Ynthe largest ofXi.Thatis,Y 1 <Y 2 <···<YnrepresentX 1 ,X 2 ,...,Xnwhen
the latter are arranged in ascending order of magnitude. We callYi,i=1, 2 ,...,n,
theithorder statisticof the random sampleX 1 ,X 2 ,...,Xn. Then the joint pdf
ofY 1 ,Y 2 ,...,Ynis given in the following theorem.
Theorem 4.4.1.Using the above notation, letY 1 <Y 2 <···<Yndenote the
norder statistics based on the random sampleX 1 ,X 2 ,...,Xnfrom a continuous
distribution with pdff(x)and support(a, b). Then the joint pdf ofY 1 ,Y 2 ,...,Ynis
given by
g(y 1 ,y 2 ,...,yn)={
n!f(y 1 )f(y 2 )···f(yn) a<y 1 <y 2 <···<yn<b
0 elsewhere. (4.4.1)Proof: Note that the support ofX 1 ,X 2 ,...,Xncan be partitioned inton! mutually
disjoint sets that map onto the support ofY 1 ,Y 2 ,...,Yn,namely,{(y 1 ,y 2 ,...,yn):
a<y 1 <y 2 <···<yn<b}.Oneofthesen!setsisa<x 1 <x 2 <···<xn<b,
and the others can be found by permuting thenxs in all possible ways. The
transformation associated with the one listed isx 1 =y 1 ,x 2 =y 2 ,...,xn=yn,
which has a Jacobian equal to 1. However, the Jacobian of each of the other
transformations is either±1. Thus
g(y 1 ,y 2 ,...,yn)=∑n!i=1|Ji|f(y 1 )f(y 2 )···f(yn)={
n!f(y 1 )f(y 2 )···f(yn) a<y 1 <y 2 <···<yn<b
0elsewhere,as was to be proved.
Example 4.4.1.LetXdenote a random variable of the continuous type with a pdf
f(x) that is positive and continuous, with supportS=(a, b),−∞ ≤a<b≤∞.
The distribution functionF(x)ofXmay be written
F(x)=∫xaf(w)dw, a<x<b.Ifx≤a, F(x) = 0; and ifb≤x, F(x) = 1. Thus there is a unique medianmof the
distribution withF(m)=^12 .LetX 1 ,X 2 ,X 3 denote a random sample from this
distribution and letY 1 <Y 2 <Y 3 denote the order statistics of the sample. Note
thatY 2 is the sample median. We compute the probability thatY 2 ≤m.Thejoint
pdfofthethreeorderstatisticsis
g(y 1 ,y 2 ,y 3 )={
6 f(y 1 )f(y 2 )f(y 3 ) a<y 1 <y 2 <y 3 <b
0elsewhere.