Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.4. Order Statistics 257

Certain functions of the order statisticsY 1 ,Y 2 ,...,Ynare important statistics
themselves. Thesample rangeof the random sample is given byYn−Y 1 and the
sample midrangeis given by (Y 1 +Yn)/2, which is called themidrangeof the
random sample. Thesample medianof the random sample is defined by

Q 2 =

{

Y(n+1)/ 2 ifnis odd

(Yn/ 2 +Y(n/2)+1)/2ifnis even.

(4.4.4)

Example 4.4.3.LetY 1 ,Y 2 ,Y 3 be the order statistics of a random sample of size
3 from a distribution having pdf

f(x)=

{

10 <x< 1

0elsewhere.

We seek the pdf of the sample rangeZ 1 =Y 3 −Y 1 .SinceF(x)=x, 0 <x<1, the
joint pdf ofY 1 andY 3 is

g 13 (y 1 ,y 3 )=

{

6(y 3 −y 1 )0<y 1 <y 3 < 1

0elsewhere.

In addition toZ 1 =Y 3 −Y 1 ,letZ 2 =Y 3. The functionsz 1 =y 3 −y 1 ,z 2 =y 3 have
respective inversesy 1 =z 2 −z 1 ,y 3 =z 2 , so that the corresponding Jacobian of the
one-to-one transformation is

J=

∣ ∣ ∣ ∣ ∣ ∣

∂y 1

∂z 1

∂y 1

∂z 2

∂y 3

∂z 1

∂y 3

∂z 2

∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣

− 11

01

∣ ∣ ∣ ∣ ∣ ∣

=− 1.

Thus the joint pdf ofZ 1 andZ 2 is

h(z 1 ,z 2 )=

{

|− 1 | 6 z 1 =6z 1 0 <z 1 <z 2 < 1

0elsewhere.

Accordingly, the pdf of the rangeZ 1 =Y 3 −Y 1 of the random sample of size 3 is

h 1 (z 1 )=

{∫

1

z 16 z^1 dz^2 =6z^1 (1−z^1 )0<z^1 <^1

0elsewhere.

4.4.1 Quantiles

LetXbe a random variable with a continuous cdfF(x). For 0<p<1, define
thepthquantileofXto beξp=F−^1 (p). For example,ξ 0. 5 , the median ofX,is
the 0.5 quantile. LetX 1 ,X 2 ,...,Xnbe a random sample from the distribution of
Xand letY 1 <Y 2 <···<Ynbe the corresponding order statistics. Letkbe the
greatest integer less than or equal to [p(n+ 1)]. We next define an estimator ofξp
after making the following observation. The area under the pdff(x)totheleftof
YkisF(Yk). The expected value of this area is

E(F(Yk)) =

∫b

a

F(yk)gk(yk)dyk,