Order Statistics
Let
12
,,,
n
XX
X
…
be a random sample from the density
()
X
fx
and suppose that we want
the joint density of
n
YY
Y
…
where the
i
Y
are the
i
X
arranged in order of magnitude
so that
n
Y
≤≤≤
…
.
In what follows, we develop the expression for the joint
density for
3
n
=
.
We assume that
X
is the density of a continuous random variable.
Let
123
,,
XXX
be a sample of size 3 from
X
.
We shall transform from the x’s to the
y’s as follows
Let
1.
11
22
33
;;
YX
===
if
{}
1
1
2
3
,, 
I
XXX X X
X
∈
=<
<
2.
23
32
if
2
1
3
2
I
X
∈
<
3.
21
if
3
2
1
3
I
XXX X
X X
∈
<
4.
31
if
4
2
3
1
I
X
X
∈
<
5.
13
if
5
3
1
2
I
∈
<
6.
if
6
3
2
1
I
X
X
∈
<
Note that
0
jk
j
k
II
I
I
•=
∩
=
(i.
.e. the regions in X space are pairwise disjoint) and
123456
1
2
3
1
2
3
,
,
IIIIII
X
X
X
X
X
X
+++++=
−∞
<<
∞−∞
∞
is the entire space.
We shall find the joint density of
YYY
for each of the above six cases or regions.
In
all cases, the joint density of
is
(
)
(
)
(
)
1
2
3
X
X
X
f
xxx
fxfxfx
=
.
Case 1
For the region
1
1
2
3
I
X
∈
<
, the Jacobian is one
and the joint density is
(
)
(
)
(
)
1
2
3
X
X
X
f
yyy
fyfyfy
=
for
and zero
elsewhere.
Case 2
For the region
2
1
3
2
I
X
∈
<
, the Jacobian is one
and the joint density is
(
)
(
)
(
)
1
3
2
X
X
X
f
=
for
and zero
elsewhere.
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View Full DocumentCase 3
For the region
()
{}
123
3
2
1
3
,,
,, 
XXX
I
XXX X
X X
∈
=<
<
, the Jacobian is one
and the joint density is
(
)
(
)
(
)
2
1
3
YYY
X
X
X
f
yyy
fyfyfy
=
for
<<
and zero
elsewhere.
Case 4
For the region
4
2
3
1
I
X
X
∈
<
, the Jacobian is one
and the joint density is
(
)
(
)
(
)
2
3
1
X
X
X
f
=
for
and zero
elsewhere.
Case 5
For the region
5
3
1
2
I
∈
<
, the Jacobian is one
and the joint density is
(
)
(
)
(
)
3
1
2
X
X
X
f
=
for
and zero
elsewhere.
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 Spring '11
 Neveskyi
 Probability distribution, probability density function, Cumulative distribution function, joint density

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