�
�
1
14.30
Introduction
to
Statistical
Methods
in
Economics
Lecture
Notes
11
Konrad
Menzel
March
17,
2009
Order Statistics
Let
X
1
,...,X
n
be independent random
variables
with identical
p.d.f.s
f
X
1
(
x
) =
...
=
f
X
n
(
x
) - we’ll
generally call
such
a
sequence ”independent and identically distributed”,
which is
typically abbreviated
as
iid
.
We
are
interested
in
the function
Y
n
= max
{
X
1
,...,X
n
}
i.e.
Y
n
is
the
largest value in
the sample.
We
can derive
the c.d.f. of
Y
n
using
independence
F
Y
n
(
y
)
=
P
(
Y
n
≤
y
) =
P
(
X
1
≤
y,X
2
≤
y,...,X
n
≤
y
)
=
P
(
X
1
≤
y
)
P
(
X
2
≤
y
)
...P
(
X
n
≤
y
)
=
F
X
1
(
y
)
F
X
2
(
y
)
...F
X
n
(
y
)
=
[
F
X
(
y
)]
n
Using the
chain rule, we can
obtain
the p.d.f. of
the maximum,
d
n
−
1
f
Y
n
(
y
) =
F
Y
n
(
y
) =
n
[
F
X
(
y
)]
f
X
(
y
)
dy
Example 1
An
old painting
is sold at an auction.
n
identical
bidders submit their bids
B
1
,...,B
n
independently,
and the
marginal c.d.f.
of
the
bids is
F
B
(
b
)
.
The
potential
buyer who
submitted
the
highest bid gets to buy the painting
and has to pay his bid (this type of auction is called Dutch, or first
price
auction).
Then the
revenue
of
the
seller
of the
painting
has p.d.f. is
n
−
1
f
Y
(
y
) =
f
max
{
B
1
,...,B
n
}
(
y
) =
n
[
F
B
(
y
)]
f
B
(
y
)
Now we
can generalize this
to
other
ranks
in
the sample,
e.g.
Y
n
−
1
=
”The 2nd
highest value in
X
1
,...,X
n
”
This
random
variable is
called the (
n
−
1)
th order statistic
of
X
1
,...,X
n
,
and we can state its
p.d.f.
Proposition 1
Let
X
1
,...,X
n
be
an iid
sequence
of random variables with
p.d.f.
f
X
(
f
)
and
c.d.f.
F
X
(
x
)
.
Then the
k
th
order statistic
Y
k
has p.d.f.
f
Y
k
(
y
) =
k
n
[
F
X
(
y
)]
k
−
1
[1
−
F
X
(
y
)]
n
−
k
f
X
(
y
)
k
1