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## Chapter 2. Order Statistics

1 The Order Statistics
For a sample of independent observations X
1
, X
2
, . . . , X
n
on a distribution F, the ordered
sample values
X
(1)
X
(2)
X
(n)
,
or, in more explicit notation,
X
(1:n)
X
(2:n)
X
(n:n)
,
are called the order statistics. If F is continuous, then with probability 1 the order statistics
of the sample take distinct values (and conversely).
There is an alternative way to visualize order statistics that, although it does not
necessarily yield simple expressions for the joint density, does allow simple derivation of many
important properties of order statistics. It can be called the quantile function representation.
The quantile function (or inverse distribution function, if you wish) is dened by
F
1
(y) = inf{x : F(x) y}. (1)
Now it is well known that if U is a Uniform(0,1) random variable, then F
1
(U) has distri-
bution function F. Moreover, if we envision U
1
, . . . , U
n
as being iid Uniform(0,1) random
variables and X
1
, . . . , X
n
as being iid random variables with common distribution F, then
(X
(1)
, . . . , X
(n)
)
d
= (F
1
(U
(1)
), . . . , F
1
(U
(n)
)), (2)
where
d
= is to be read as has the same distribution as.
1.1 The Quantiles and Sample Quantiles
Let F be a distribution function (continuous from the right, as usual). The proof of F is
right continuous can be obtained from the following fact:
F(x +h
n
) F(x) = P(x < X x +h
n
),
where {h
n
} is a sequence of real numbers such that 0 < h
n
0 as n . It follows from
the continuity property of probabaility (P(lim
n
A
n
) = lim
n
P(A
n
) if limA
n
exists.) that
lim
n
[F(x +h
n
) F(x)] = 0,
and hence that F is right-continuous. Let D be the set of all discontinuity points of F and
n be a positive integer. Set
D
n
=
_
x D : P(X = x)
1
n
_
.
2
Since F()F() = 1, the number of elements in D
n
cannot exceed n. Clearly D =
n
D
n
,
and it follows that D is countable. Or, the set of discontinuity points of a distribution
function F is countable. We then conclude that every distribution function F admits the
decomposition
F(x) = F
d
(x) + (1 )F
c
(x), (0 1),
where F
d
and F
c
are both continuous function such that F
d
is a step function and F
c
is
continuous. Moreover, the above decomposition is unique.
Let denote the Lebesgue measure on B, the -eld of Borel sets in R. It follows from
the Lebesgue decomposition theorem that we can write F
c
(x) = F
s
(x)+(1)F
ac
(x) where
0 1, F
s
is singular with respect to , and F
ac
is absolutely continuous with respect to
. On the other hand, the Radon-Nikodym theorem implies that there exists a nonnegative
Borel-measurable function on R such that
F
ac
(x) =
_
x

fd,
where f is called the Radon-Nikodym derivative. This says that every distribution function
F(x) =
1
F
d
(x) +
2
F
s
(x) +
3
F
ac
(x), (x R),
where
i
0 and

3
i=1

i
= 1.
For 0 < p < 1, the pth quantile or fractile of F is dened as
(p) = F
1
(p) = inf{x : F(x) p}.
This denition is motivated by the following observation:
If F is continuous and strictly increasing, F
1
is dened by
F
1
(y) = x when y = F(x).
If F has a discontinuity at x
0
, suppose that F(x
0
) < y < F(x
0
) = F(x
0
+). In this
case, although there exists no x for which y = F(x), F
1
(y) is dened to be equal to
x
0
.
Now consider the case that F is not strictly increasing. Suppose that
F(x)
_

_
< y for x < a
= y for a x b
> y for x > b
Then any value a x b could be chosen for x = F
1
(y). The convention in this case
is to dene F
1
(y) = a.
3
Now we prove that if U is uniformly distributed over the interval (0, 1), then X =
F
1
X
(U) has cumulative distribution function F
X
(x). The proof is straightforward:
P(X x) = P[F
1
X
(U) x] = P[U F
X
(x)] = F
X
(x).
Note that discontinuities of F become converted into at stretches of F
1
and at stretches
of F into discontinuities of F
1
.
In particular,
1/2
= F
1
(1/2) is called the median of F. Note that
p
satises
F((p)) p F((p)).
The function F
1
(t), 0 < t < 1, is called the inverse function of F. The following
proposition, giving useful properties of F and F
1
, is easily checked.
Lemma 1 Let F be a distribution function. The function F
1
(t), 0 < t < 1, is nondecreasing
and left-continuous, and satises
(i) F
1
(F(x)) x, < x < ,
(ii) F(F
1
(t)) t, 0 < t < 1.
Hence
(iii) F(x) t if and only if x F
1
(t).
Corresponding to a sample {X
1
, X
2
, . . . , X
n
} of observations on F, the sample pth
quantile is dened as the pth quantile of the sample distribution function F
n
, that is, as
F
1
n
(p). Regarding the sample pth quantile as an estimator of
p
, we denote it by

pn
, or
simply by

p
when convenient.
Since the order stastistics is equivalent to the sample distribution function F
n
, its role
is fundamental even if not always explicit. Thus, for example, the sample mean may be
regarded as the mean of the order statistics, and the sample pth quantile may be expressed
as

pn
=
_
_
_
X
n,np
if np is an integer
X
n,[np]+1
if np is not an integer.
1.2 Functions of Order Statistics
Here we consider statistics which may be expressed as functions of order statistics. A variety
of short-cut procedures for quick estimates of location or scale parameters, or for quick tests
of related hypotheses, are provided in the form of linear functions of order statistics, that is
statistics of the form
n

i=1
c
ni
X
(i:n)
.
4
We term such statistics L-estimates. For example, the sample range X
(n:n)
X
(1:n)
belongs
to this class. Another example is given by the -trimmed mean.
1
n 2[n]
n[n]

i=[n]+1
X
(i:n)
,
which is a popular competitor of

X for robust estimation of location. The asymptotic dis-
tribution theory of L-statistics takes quite dierent forms, depending on the character of the
coecients {c
ni
}.
The representations of

X and

pn
in terms of order statistics are a bit articial. On
the other hand, for many useful statistics, the most natural and ecient representations are
in terms of order statistics. Examples are the extreme values X
1:n
and X
n:n
and the sample
range X
n:n
X
1:n
.
1.3 General Properties
Theorem 1 (1) P(X
(k)
x) =

n
i=k
C(n, i)[F(x)]
i
[1 F(x)]
ni
for < x < .
(2) The density of X
(k)
is given by nC(n 1, k 1)F
k1
(x)[1 F(x)]
nk
f(x).
(3) The joint density of X
(k
1
)
and X
(k
2
)
is given by
n!
(k
1
1)!(k
2
k
1
1)!(n k
2
)!
[F(x
(k
1
)
)]
k
1
1
[F(x
(k
2
)
) F(x
(k
1
)
)]
k
2
k
1
1
[1 F(x
(k
2
)
)]
nk
2
f(x
(k
1
)
)f(x
(k
2
)
)
for k
1
< k
2
and x
(k
1
)
< x
(k
2
)
.
(4) The joint pdf of all the order statistics is n!f(z
1
)f(z
2
) f(z
n
) for < z
1
< <
z
n
< .
(5) Dene V = F(X). Then V is uniformly distributed over (0, 1).
Proof. (1) The event {X
(k)
x} occurs if and only if at least k out of X
1
, X
2
, . . . , X
n
are
less than or equal to x.
(2) The density of X
(k)
is given by nC(n 1, k 1)F
k1
(x)[1 F(x)]
nk
f(x). It can be
shown by the fact that
d
dp
n

i=k
C(n, i)p
i
(1 p)
ni
= nC(n 1, k 1)p
k1
(1 p)
nk
.
Heuristically, k 1 smallest observations are x and n k largest are > x. X
(k)
falls into a
small interval of length dx about x is f(x)dx.
5
1.4 Conditional Distribution of Order Statistics
In the following two theorems, we relate the conditional distribution of order statistics (con-
ditioned on another order statistic) to the distribution of order statistics from a population
whose distribution is a truncated form of the original population distribution function F(x).
Theorem 2 Let X
1
, X
2
, . . . , X
n
be a random sample from an absolutely continuous popula-
tion with cdf F(x) and density function f(x), and let X
(1:n)
X
(2:n)
X
(n:n)
denote
the order statistics obtained from this sample. Then the conditional distribution of X
(j:n)
,
given that X
(i:n)
= x
i
for i < j, is the same as the distribution of the (j i)th order statistic
obtained from a sample of size n i from a population whose distribution is simply F(x)
truncated on the left at x
i
.
Proof. From the marginal density function of X
(i:n)
and the joint density function of
X
(i:n)
and X
(j:n)
, we have the conditional density function of X
(j:n)
, given that X
(i:n)
= x
i
,
as
f
X
(j:n)
(x
j
|X
(i:n)
= x
i
) = f
X
(i:n)
,X
(j:n)
(x
i
, x
j
)/f
X
(i:n)
(x
i
)
=
(n i)!
(j i 1)!(n j)!
_
F(x
j
) F(x
i
)
1 F(x
i
)
_
ji1

_
1 F(x
j
)
1 F(x
i
)
_
nj
f(x
j
)
1 F(x
i
)
.
Here i < j n and x
i
x
j
< . The result follows easily by realizing that {F(x
j
)
F(x
i
)}/{1F(x
i
)} and f(x
j
)/{1F(x
i
)} are the cdf and density function of the population
whose distribution is obtained by truncating the distribution F(x) on the left at x
i
.
Theorem 3 Let X
1
, X
2
, . . . , X
n
be a random sample from an absolutely continuous popula-
tion with cdf F(x) and density function f(x), and let X
(1:n)
X
(2:n)
X
(n:n)
denote
the order statistics obtained from this sample. Then the conditional distribution of X
(i:n)
,
given that X
(j:n)
= x
j
for j > i, is the same as the distribution of the ith order statistic in
a sample of size j 1 from a population whose distribution is simply F(x) truncated on the
right at x
j
.
Proof. From the marginal density function of X
(i:n)
and the joint density function of X
(i:n)
and X
(j:n)
, we have the conditional density function of X
(i:n)
, given that X
(j:n)
= x
j
, as
f
X
(i:n)
(x
i
|X
(j:n)
= x
j
) = f
X
(i:n)
,X
(j:n)
(x
i
, x
j
)/f
X
(j:n)
(x
j
)
=
(j 1)!
(i 1)!(j i 1)!
_
F(x
i
)
F(x
j
)
_
i1

_
F(x
j
) F(x
i
)
F(x
j
)
_
ji1
f(x
i
)
F(x
j
)
.
6
Here 1 i < j and < x
i
x
j
. The proof is completed by noting that F(x
i
)/F(x
j
) and
f(x
i
)/F(x
j
) are the cdf and density function of the population whose distribution is obtained
by truncating the distribution F(x) on the right at x
j
1.5 Computer Simulation of Order Statistics
In this section, we will discuss some methods of simulating order statistics from a distribution
F(x). First of all, it should be mentioned that a straightforward way of simulating order
statistics is to generate a pseudorandom sample from the distribution F(x) and then sort
the sample through an ecient algorithm like quick-sort. This general method (being time-
consuming and expensive) may be avoided in many instances by making use of some of the
distributional properties to be established now.
For example, if we wish to generate the complete sample (x
(1)
, . . . , x
(n)
) or even a Type
II censored sample (x
(1)
, . . . , x
(r)
) from the standard exponential distribution. This may be
done simply by generating a pseudorandom sample y
1
, . . . , y
r
from the standard exponential
distribution rst, and then setting
x
(i)
=
i

j=1
y
j
/(n j + 1), i = 1, 2, . . . , r.
The reason is as follows:
Theorem 4 Let X
(1)
X
(2)
X
(n)
be the order statistics from the standard exponen-
tial distribution. Then, the random variables Z
1
, Z
2
, . . . , Z
n
, where
Z
i
= (n i + 1)(X
(i)
X
(i1)
),
with X
(0)
0, are statistically independent and also have standard exponential distributions.
Proof. Note that the joint density function of X
(1)
, X
(2)
, . . . , X
(n)
is
f
1,2,...,n:n
(x
1
, x
2
, . . . , x
n
) = n! exp
_

i=1
x
i
_
, 0 x
1
< x
2
< < x
n
< .
Now let us consider the transformation
Z
1
= nX
(1)
, Z
2
= (n 1)(X
(2)
X
(1)
), . . . , Z
n
= X
(n)
X
(n1)
,
or the equivalent transformation
X
(1)
= Z
1
/n, X
(2)
=
Z
1
n
+
Z
2
n 1
, . . . , X
(n)
=
Z
1
n
+
Z
2
n 1
+ +Z
n
.
After noting the Jacobian of this transformation is 1/n! and that

n
i=1
x
i
=

n
i=1
z
i
, we
immediately obtain the joint density function of Z
1
, Z
2
, . . . , Z
n
to be
f
Z
1
,Z
2
,...,Z
n
(z
1
, z
2
, . . . , z
n
) = exp
_

i=1
z
i
_
, 0 z
1
, . . . , z
n
< .
7
If we wish to generate order statistics from the Uniform(0, 1) distribution, we may
use the following two Theorems and avoid sorting once again. For example, if we only need
the ith order statistic u
(i)
, it may simply be generated as a pseudorandom observation from
Beta(i, n i + 1) distribution.
Theorem 5 For the Uniform(0, 1) distribution, the random variables V
1
= U
(i)
/U
(j)
and
V
2
= U
(j)
, 1 i < j n, are statistically independent, with V
1
and V
2
having Beta(i, j i)
and Beta(j, n j + 1) distributions, respectively.
Proof. From Theorem 1(3), we have the joint density function of U
(i)
and U
(j)
(1 i < j
n) to be
f
i,j:n
(u
i
, u
j
) =
n!
(i 1)!(j i 1)!(n j)!
u
i1
i
(u
j
u
i
)
ji1
(1 u
j
)
nj
, 0 < u
i
< u
j
< 1.
Now upon makin the transformation V
1
= U
(i)
/U
(j)
and V
2
= U
(j)
and noting that the
Jacobian of this transformation is v
2
, we derive the joint density function of V
1
and V
2
to be
f
V
1
,V
2
(v
1
, v
2
) =
(j 1)!
(i 1)!(j i 1)!
v
i1
1
(1 v
1
)
ji1
=
n!
(j 1)!(n j)!
v
j1
2
(1 v
2
)
nj
,
0 < v
1
< 1, 0 < v
2
< 1. From the above equation it is clear that the random variables V
1
and V
2
are statistically independent, and also that they are distributed as Beta(i, j i) and
Beta(j, n j + 1), respectively.
Theorem 6 For the Uniform(0, 1) distribution, the random variables
V

1
=
U
(1)
U
(2)
, V

2
=
_
U
(2)
U
(3)
_
2
, , V

(n1)
=
_
U
(n1)
U
(n)
_
n1
and V

n
= U
n
(n)
are all independent Uniform(0, 1) random variables.
Proof. Let X
(1)
< X
(2)
< < X
(n)
denote the order statistics from the standard expo-
nential distribution. Then upon making use of the facts that X = log U has a standard
exponential distribution and that log u is a monotonically decreasing function in u, we
immediately have X
(i)
d
= log U
(ni+1)
. The above equation yields
V

i
=
_
U
(i)
U
(i+1)
_
I
d
=
_
e
X
(ni+1)
e
X
(ni)
_
= exp[i(X
(ni+1)
X
(ni)
)]
d
= exp(Y
ni+1
)
upon using the above theorem, where Y
,
i
are independent standard exponential random vari-
ables.
The just-described methods of simulating uniform order statistics may also be used
easily to generate order statistics from any known distribution F(x) for which F
1
() is
relatively easy to compute. We may simply obtain the order statistics x
(1)
, . . . , x
(n)
from the
required distribution F() by setting x
(i)
= F
1
(u
(i)
).
8
2 Large Sample Properties of Sample Quantile
2.1 An Elementary Proof
Consider the sample pth quantile,

pn
, which is X
([np])
or X
([np]+1)
depending on whether
np is an integer (here [np] denotes the integer part of np). For simplicity, we discuss the
properties of X
([np])
where p (0, 1) and n is large. This will in turn inform us of the
properties of

pn
.
We rst consider the case that X is uniformly distributed over [0, 1]. Let U
([np])
denote
the sample pth quantile. If i = [np], we have
nC(n 1, i 1) =
n!
(i 1)!(n i)!
=
(n + 1)
(i)(n i + 1)
= B(i, n i + 1).
Elementary computations beginning with Theorem 1(2) yields U
([np])
Beta(i
0
, n + 1 i
0
)
where i
0
= [np]. Then U
(i
0
)
Beta(i
0
, n i
0
+ 1). Note that
EU
([np])
=
[np]
n + 1
p
nCov
_
U
(np
1
)
, U
(np
2
)
_
= n
[np
1
](n + 1 [np
2
])
(n + 1)
2
(n + 2)
p
1
(1 p
2
).
Use these facts and Chebyschev inequality, we can show easily that U
([np])
P
p with rate
n
1/2
. This generates the question whether we can claim that

pn
P

p
.
Recall that U = F(X). If F is absolutely continuous with nite positive density f at
p
, it is
expected that the above claim holds.
Recall that U
([np])
P
p with rate n
1/2
. The next question would be what the distri-
bution of

n(U
([np])
p) is? Note that U
([np])
is a Beta([np], n [np] + 1) random variable.
Thus, it can be expressed as
U
[np]
=

i
0
i=1
V
i

i
0
i=1
V
i
+

n+1
i=i
0
+1
V
i
,
where the V
i
s are iid Exp(1) random variables. Observe that

n
_

[np]
i=1
V
i

[np]
i=1
V
i
+

n+1
i=[np]+1
V
i
p
_
=
1

n
_
(1 p)
_

[np]
i=1
V
i
[np]
_
p
_

n+1
[np]+1
V
i
(n [np] + 1)
_
+ [(1 p)[np] p(n [np] + 1)]
_

n+1
i=1
V
i
/n
and (

n)
1
{(1 p)[np] p(n [np] + 1)} 0. Since E(V
i
) = 1 and V ar(V
i
) = 1, from the
central limit theorem it follows that

i
0
i=1
V
i
i
0

i
0
d
N (0, 1)
9
and

n+1
i=i
0
+1
V
i
(n i
0
+ 1)

n i
0
+ 1
d
N (0, 1) .
Consequently,

i
0
i=1
V
i
i
0

n
d
N (0, p)
and

n+1
i=i
0
+1
V
i
(n i
0
+ 1)

n
d
N (0, 1 p) .
Since

i
0
i=1
V
i
and

n
i=i
0
+1
V
i
are independent for all n,
(1 p)

[np]
i=1
V
i
[np]

n
p

n+1
[np]+1
V
i
(n [np] + 1)

n
d
N(0, p(1 p)).
Now, using the weak law of large numbers, we have
1
n + 1
n+1

i=1
V
i
P
1.
Hence, by Slutskys Theorem,

n
_
U
([np])
p
_
d
N (0, p(1 p)) .
For an arbitrary cdf F, we have X
([np])
= F
1
(U
([np])
). Upon expanding F
1
(U
([np])
)
in a Taylor-series around the point E(U
([np])
) = [np]/(n + 1), we get
X
([np])
d
= F
1
(p) + (U
([np])
p){f(F
1
(D
n
))}
1
where the random variable D
n
is betwen U
([np])
and p. This can be rearranged as

n
_
X
([np])
F
1
(p)
_
d
=

n
_
U
([np])
p
_
{f(F
1
(D
n
)}
1
.
When f is continuous at F
1
(p), it follows that as n , f(F
1
(D
n
))
P
f(F
1
(p)). Use
the delta method, it yields

n
_
X
([np])
F
1
(p)
_
d
N
_
0,
p(1 p)
[f(F
1
(p))]
2
_
.
2.2 A Probability Inequality for |

p
|
We shall use the following result of Hoeding (1963) to show that P(|

pn

p
| > ) 0
exponentially fast.
Lemma 2 Let Y
1
, . . . , Y
n
be independent random variables satisfying P(a Y
i
b) = 1,
each i, where a < b. Then, for t > 0,
P
_
n

i=1
Y
i

i=1
E(Y
i
) nt
_
exp
_
2nt
2
/(b a)
2

.
10
Remark. Suppose that Y
1
, Y
2
, . . . , Y
n
are independent and identically distributed random
variables. Use Bery-Esseen theorem, we have
P
_
n

i=1
Y
i

i=1
E(Y
i
) nt
_
= (t
_
V ar(Y
1
)/

n) +Error.
Here
|Error|
c

n
E|Y
1
EY
1
|
3
[V ar(Y
1
)]
3/2
.
Theorem 7 Let 0 < p < 1. Suppose that
p
is the unique solution x of F(x) p F(x).
Then, for every > 0,
P
_
|

pn

p
| >
_
2 exp
_
2n
2

_
, all n,
where

= min{F(
p
+) p, p F(
p
)}.
Proof. Let > 0. Write
P
_
|

pn

p
| >
_
= P
_

pn
>
p
+
_
+P
_

pn
<
p

_
.
By Lemma ??,
P
_

pn
>
p
+
_
= P(p > F
n
(
p
+))
= P
_
n

i=1
I(X
i
>
p
+) > n(1 p)
_
= P
_
n

i=1
V
i

i=1
E(V
i
) > n
1
_
,
where V
i
= I(X
i
>
p
+) and
1
= F(
p
+) p. Likewise,
P
_

pn
<
p

_
= P(p > F
n
(
p
))
= P
_
n

i=1
W
i

i=1
E(W
i
) > n
2
_
,
where W
i
= I(X
i
<
p
) and
2
= p F(
p
). Therefore, utilizing Hoedings lemma,
we have
P
_

pn
>
p
+
_
exp
_
2n
2
1
_
and
P
_

pn
<
p

_
exp
_
2n
2
2
_
.
Putting

= min{
1
,
2
}, the proof is completed.
11
2.3 Asymptotic Normality
Theorem 8 Let 0 < p < 1. Suppose that F possesses a density f in a neighborhood of
p
and f is positive and continuous at
p
, then

n(

pn

p
)
d
N
_
0,
p(1 p)
[f(
p
)]
2
_
.
Proof. We only consider p = 1/2. Note that
p
is the unique median since f(
p
) > 0.
First, we consider that n is odd (i.e., n = 2m1).
P
_

n
_
X
(m)
F
1
_
1
2
__
t
_
= P
_

nX
(m)
t

F
1
_
1
2
_
= 0
_
= P
_
X
(m)
t/

F
1
_
1
2
_
= 0
_
.
Let S
n
be the number of Xs that exceed t/

n. Then
X
(m)

t

n
if and only if S
n
m1 =
n 1
2
.
Or, S
n
is a Bin(n, 1 F(F
1
(1/2) + tn
1/2
)) random variable. Set F
1
(1/2) = 0 and
p
n
= 1 F(n
1/2
t). Note that
P
_

n
_
X
(m)
F
1
_
1
2
__
t
_
= P
_
S
n

n 1
2
_
= P
_
S
n
np
n
_
np
n
(1 p
n
)

1
2
(n 1) np
n
_
np
n
(1 p
n
)
_
.
Recall that p
n
depends on n. Write
S
n
np
n
_
np
n
(1 p
n
)
=
n

i=1
Y
i
p
n
_
np
n
(1 p
n
)
=
n

i=1
Y
in
.
Again, they can be expressed as a double array with Y
i
Bin(1, p
n
)
Now utilize the Berry-Esseen Theorem to have
P
_
S
n

n 1
2
_

_
1
2
(n 1) np
n
_
np
n
(1 p
n
)
_
0
as n . Writing
1
2
(n 1) np
n
_
np
n
(1 p
n
)

n
_
1
2
p
n
_
1/2
=

n
_

1
2
+F(n
1/2
t)
_
1/2
= 2t
F(n
1/2
t) F(0)
n
1/2
t
2tf(0).
Thus,

_
1
2
(n 1) np
n
_
np
n
(1 p
n
)
_
(2f(0) t)
or

n
_
X
(m)
F
1
(1/2)
_
d
N
_
0,
1
4f
2
(F
1
(1/2))
_
.
12
When n is even (i.e., n = 2m), both P[

n(X
(m)
F
1
(1/2)) t] and P[

n(X
(m+1)

F
1
(1/2)) t] tend to
_
2f(F
1
(1/2)) t
_
.
We just prove asymptotic normality of

p
in the case that F possesses derivative at
the point
p
. So far we have discussed in detail the asymptotic normality of a single quantile.
This discussion extends in a natural manner to the asymptotic joint normality of a xed
number of quantiles. This is made precise in the following result.
Theorem 9 Let 0 < p
1
< < p
k
< 1. Suppose that F has a density f in neighborhoods
of
p
1
, . . . ,
p
k
and that f is positive and continuous at
p
1
, . . . ,
p
k
. Then (

p
1
, . . . ,

p
k
) is
asymptotically normal with mean vector (
p
1
, . . . ,
p
k
) and covariance
ij
/n, where

ij
=
p
i
(1 p
j
)
f(
p
i
)f(
p
j
)
for 1 i j k
and
ij
=
ji
for i > j.
Suppose that we have a sample of size n from a normal distribution N(,
2
). Let m
n
represent the median of this sample. Then because f() = (

2)
1
,

n(m
n
)
d
N(0, (1/4)/f
2
()) = N(0,
2
/2).
Compare m
n
with

X
n
as an estimator of . We conclude immediately that

X
n
is better than
m
n
since the latter has a much larger variance. Now consider the above problem again with
Cauchy distribution C(, ) with density function
f(x) =
1

1
1 + [(x )/]
2
.
2.4 A Measure of Dispersion Based on Quantiles
The joint normality of a xed number of central order statistics can be used to construct
simultaneous condence regions for two or more population quantiles. As an illustration, we
now consider the semi-interquantile range, R =
1
2
(
3/4

1/4
). (Note that the parameter
in C(, ) is the semi-interquantile range.) It is used as an alternative to to measure the
spread of the data. A natural estimate of R is

R
n
=
1
2
(

3/4

1/4
). Theorem 4 gives the joint
distribution of

1/4
and

3/4
. We can use the following result, due to Cramer and Wold (1936),
which reduces the convergence of multivariate distribution functions to the convergence of
univariate distribution functions.
Theorem 10 In R
k
, the random vectors X
n
converge in distribution to the random vector
X if and only if each linear combination of the components of X
n
converges in distribution
to the same linear combination of the components of X.
13
By Theorem 9 and the Cramer-Wold device, we have

R AN
_
R,
1
64n
_
3
[f(
1/4
)]
2

2
f(
1/4
)f(
3/4
)
+
3
[f(
3/4
)]
2
__
.
For F = N(,
2
), we have

R AN
_
0.6745,
(0.7867)
2

2
n
_
.
2.5 Condence Intervals for Population Quantiles
Assume that F posseses a density f in a neighborhood of
p
and f is positive and continuous
at
p
. For simplicity, consider p = 1/2. Then

n
_

1/2

1/2
_
d
N
_
0,
1
4f
2
(
1/2
)
_
.
Therefore, we can derive a condence interval of
1/2
if either f(
1/2
) is known or a good
estimator of f(
1/2
) is available. A natural question to ask then is how do we estimate f(
1/2
).
Here we propose two estimates. The rst estimate is
number of observatiobns falling in
_

1/2
h
n
,
1/2
+h
n
_
2h
n
which is motivated by
F
_

1/2
+h
n
_
F
_

1/2
h
n
_
2h
n
f(
1/2
).
The second one is
X
[n/2+]
X
[n/2]
2/n
where = O(n
d
) for 0 < d < 1.
2.6 Distribution-Free Condence Interval
When F is absolutely continuous, F(F
1
(p)) = p and, hence, we have
P(X
(i:n)
F
1
(p)) = P(F(X
(i:n)
) p) = P(U
(i:n)
p)
=
n

r=i
C(n, r)p
r
(1 p)
nr
. (3)
Now for i < j, consider
P(X
(i:n)
F
1
(p)) = P(X
(i:n)
F
1
(p), X
(j:n)
< F
1
(p))
+P(X
(i:n)
F
1
(p), X
(j:n)
F
1
(p))
= P(X
(j:n)
< F
1
(p)) +P(X
(i:n)
F
1
(p) X
(j:n)
).
Since X
(j:n)
is absolutely continuous, this equation can be written as
P
_
X
(i:n)
F
1
(p) X
(j:n)
_
= P
_
X
(i:n)
F
1
(p)
_
P
_
X
(j:n)
F
1
(p)
_
=
j1

r=i
C(n, r)p
r
(1 p)
nr
, (4)
14
where the last equality follows from (??). Thus, we have a condence interval [X
(i:n)
, X
(j:n)
]
for F
1
(p) whose condence coecient (i, j) given by (??), is free of F and can be read from
the table of binomial probabilities. If p and the desired condence level
0
are specied, we
choose i and j so that (i, j) exceeds
0
. Because of the fact that (i, j) is a step function,
usually the interval we obtain tends to be conservative. Further, the choice of i and j is not
unique, and the choice which makes (j i) small appear reasonable. For a given n and p,
the binomial pmf C(n, r)p
r
(1 p)
nr
increases as r increases up to around [np], and then
decreases. So if we want to make (j i) small, we have to start with i and j close to [np]
and gradually increase (j i) until (i, j) exceeds
0
.
2.7 Q-Q Plot
Wilk and Gnanadesikan (1968) proposed a graphical, rather informal, method of testing the
goodness-of-t of a hypothesized distribution to given data. It essentially plots the quantile
function of one cdf against that of another cdf. When the latter cdf is the empirical cdf
dened below, order statistics come into the picture. The empirical cdf, to be denoted by
F
n
(x) for all real x, represents the proportion of sample values that do not exceed x. It has
jumps of magnitude 1/n at X
(i:n)
, 1 i n. Thus, the order statistics represent the values
taken by F
1
n
(p), the sample quantile function.
The Q-Q plot is the graphical representation of the points (F
1
(p), X
(i:n)
), where
population quantiles are recorded along the horizontal axis and the sample quantiles on the
vertical axis. If the sample is in fact from F, we expect the Q-Q plot to be close to a straight
line. If not, the plot may show nonlinearity at the upper or lower ends, which may be an
indication of the presence of outliers. If the nonlinearity shows up at other points as well,
one could question the validity of the assumption that the parent cdf is F.
3 Empirical Distribution Function
Consider an i.i.d. sequence {X
i
} with distribution function F. For each sample of size n,
{X
1
, . . . , X
n
}, a corresponding sample (empirical) distribution function F
n
is constructed by
placing at each observation X
i
a mass 1/n. Thus F
n
may be represented as
F
n
(x) =
1
n
n

i=1
I(X
i
x), < x < .
(The denition of F dened on R
k
is completely analogous.)
For each xed sample, {X
1
, . . . , X
n
}, F
n
() is a distribution function, considered as
a function of x. On the other hand, for each xed value of x, F
n
(x) is a random variable,
considered as a function of the sample. In a view encompassing both features, F
n
() is a
15
random distribution function and thus may be treated as a particular stochastic process (a
random element of a suitable space.)
Note that the exact distribution of nF
n
(x) is simply binomial(n, F(x)). It follows
immediately that
Theorem 11 1. E[F
n
(x)] = F(x).
2. V ar{F
n
(x)} =
F(x)[1F(x)]
n
0, as n .
3. For each xed x, < x < ,
F
n
(x) is AN
_
F(x),
F(x)[1 F(x)]
n
_
.
The sample distribution function is utilized in statistical inference in several ways.
Firstly, its most direct application is for estimation of the population distribution function
F. Besides pointwise estimation of F(x), each x, it is also of interest to characterize globally
the estimation of F by F
n
. For each xed x, the strong law of large numbers implies that
F
n
(x)
a.s.
F(x).
We now describe the Glivenko-Cantelli Theorem which ensures that the ecdf converges uni-
formly almost surely to the true distribution function.
Theorem 12
P{sup
x
|F
n
(x) F(x)| 0} = 1.
Proof. Let > 0. Find an integer k > 1/ and numbers
= x
0
< x
1
x
2
x
k1
< x
k
= ,
such that F(x

j
) j/k F(x
j
) for j = 1, . . . , k 1. Note that if x
j1
< x
j
, then F(x

j
)
F(x
j1
) . From the strong law of large numbers,
F
n
(x
j
)
a.s.
F(x
j
) and F
n
(x

j
)
a.s.
F(x

j
)
for j = 1, . . . , k 1. Hence,

n
= max(|F
n
(x
j
) F(x
j
)|, |F
n
(x

j
) F(x

j
)|, j = 1, . . . , k 1)
a.s.
0.
Let x be arbitrary and nd j such that x
j1
< x x
j
. Then,
F
n
(x) F(x) F
n
(x

j
) F(x
j1
) F
n
(x

j
) F(x

j
) +,
and
F
n
(x) F(x) F
n
(x
j1
) F(x

j
) F
n
(x
j1
) F(x
j1
) .
This implies that
sup
x
|F
n
(x) F(x)|
n
+
a.s.
.
Since this holds for all > 0, the theorem follows.
16
3.1 Kolmogorov-Smirnov Test
To this eect, a very useful measure of closeness of F
n
to F is the Kolmogorov-Smirnov
distance
D
n
= sup
<x<
|F
n
(x) F
0
(x)|.
A related problem is to express condence bands for F(x), < x < . Thus, for selected
functions a(x) and b(x), it is of interest to compute probabilities of the form
P(F
n
(x) a(x) F(x) F
n
(x) +b(x), < x < ).
The general problem is quite dicult. However, in the simplest case, namely a(x) = b(x) = d,
the problem reduces to computation of P(D
n
< d).
For the case of F 1-dimensional, an exponential-type probability inequality for D
n
was
established by Dvoretzky, Kiefer, and Wolfowitz (1956).
Theorem 13 The distribution of D
n
under H
0
is the same for all continuous distribution.
Proof. For simplicity we give the proof for F
0
strictly increasing. Then F
0
has inverse F
1
0
and as u ranges over (0, 1), F
1
0
(u) ranges over a;; the possible values of X. Thus
D
n
= sup
0<u<1
|F
n
(F
1
0
(u))F
0
(F
1
0
(u))| = sup
0<u<1
|F
n
(F
1
0
(u)) u|.
Next note that
F
n
(F
1
0
(u)) = [number of X
i
F
1
0
(u)]/n = [number of F
0
(X
i
) u]/n.
Let U
i
= F
0
(X
i
)|. Then U
1
, . . . , U
n
are a sample from UNIF(0, 1), since
P[F
0
(X
i
) u] = P[X
i
F
1
0
(u)] = F
0
(F
1
0
(u)) = u, 0 < u < 1.
Thus,
D
n
= sup
0<u<1
|F

n
(u) u|
where F

n
(u) is the empirical distribution of the uniform sample U
1
, . . . , U
n
and the distribu-
tion of D
n
does not depend on F
0
.
We now give an important fact which is used in Donsker (1952) to give a rigorous proof
of the Kolmogorov-Smirnov Theorems.
Theorem 14 The distribution of the order statistic (Y
(1)
, . . . , Y
(n)
) of n iid random variables
Y
1
, Y
2
, . . . from the uniform distribution on [0, 1] can also be obtained as the distribution of
the ratios
_
S
1
S
n+1
,
S
2
S
n+1
, ,
S
n
S
n+1
_
,
where S
k
= T
1
+ +T
k
, k 1, and T
1
, T
2
, . . . is an iid sequence of (mean 1) exponentially
distributed random variables.
17
Intuitively, if the T
i
are regarded as the successive times between occurrence of some phenom-
ena, then S
n+1
is the time to the (n + 1)st occurrence and, in units of S
n+1
, the occurrence
times should be randomly distributed because of lack of memory and independence properties.
Recall that D
n
= sup
0<u<1
|F

n
(u) u| where F

n
(u) is the empirical distribution of
the uniform sample U
1
, . . . , U
n
. We then have
D
n
=

n sup
0<u<1
|F

n
(u) u| =

nmax
kn

Y
(k)

k
n

d
=

nmax
kn

S
k
S
n+1

k
n

=
n
S
n+1
max
kn

S
k
k

n

k
n
S
n+1
n

Theorem 15 Let F be dened on R. There exists a nite positive constant C (not depending
on F) such that
P(D
n
< d) C exp(2nd
2
), d > 0,
for all n = 1, 2, . . ..
Moreover,
Theorem 16 Let F be 1-dimensional and continuous.. Then
lim
n
P(n
1/2
D
n
< d) = 1 2

j=1
(1)
j+1
exp(2j
2
d
2
), d > 0,
for all n = 1, 2, . . ..
Secondly, we consider goodness of t test statistics based on the sample distribution function.
The null hypotheseis in the simple case is H
0
: F = F
0
, where F
0
is specied. A useful
procedure is the Kolmogorov-Smirnov test statistic

n
= sup
<x<
|F
n
(x) F
0
(x)|,
which reduces to D
n
under the null hypothesis. More broadly, a class of such statistics is
obtained by introducing weight functions:
sup
<x<
|w(x)[F
n
(x) F
0
(x)]|.
Another important class of statistics is based on the Cramer-von Mises test statistic
C
n
= n
_

[F
n
(x) F
0
(x)]
2
dF
0
(x)
and takes the general form n
_
w(F
0
(x))[F
n
(x) F
0
(x)]
2
dF
0
(x). For example, for w(t) =
[t(1t)]
1
, each duscrepancy F
n
(x)F(x) becomes weighted by the reciprocal of its standard
deviation (under H
0
), yielding the Anderson-Darling statistic.
18
3.2 Stieltjes Integral
If [a, b] is a compact interval, a set of points {x
0
, x
1
, . . . , x
n
}, satisfying the inequalities
a = x
0
< x
1
< < x
n
= b,
is called a partition of [a, b]. Write f
k
= f(x
k
) f(x
k1
) for k = 1, 2 . . . , n. If there exists
a positive number M such that
n

k=1
| f
k
| M
for all partitions of [a, b], then f is said to be of bounded variation on [a, b]. Let F(x) be a
function of bounded variation and continuous from the left such as a distribution function.
Given a nite interval (a, b) and a function f(x) we can form the sum
J
n
=
n

i=1
f(x

i
)[F(x
i
) F(x
i1
)]
for a division of (a, b) by points x
i
such that a < x
1
< < x
n
< b and arbitrary x

i

(x
i1
, x
i
). It may be noted that in the Riemann integration a similar sum is considered with
the length of the interval (x
i
x
i1
i
) F(x
i1
). If J = lim
n
J
n
as the
length of each interval 0, then J is called the Stieltjes integral of f(x) with respect to F(x)
and is denoted by
J =
_
b
a
f(x)dF(x).
The improper integral is dened by
lim
a,b
_
b
a
f(x)dF(x) =
_
f(x)dF(x).
One point of departure from Riemann integration is that it is necessary to specify
whether the end points are included in the integration or not. From the denition it is easily
shown that
_
b
a
f(x)dF(x) =
_
b
a
+
0
f(x)dF(x) +f(a)[F(a
+
0) F(a)]
where a
+
0 indicates that the end point a is not included. If F(x) jumps at a, then
_
b
a
f(x)dF(x)
_
b
a
+
0
f(x)dF(x) +f(a)[F(a
+
0) F(a)],
so that the integral taken over an interval that reduces to zero need not be zero. We shall
follow the convention that the lower end point is always included but not the upper end point.
With this convention, we see that
_
b
a
dF(x) = F(b) F(a). If there exists a function p(x)
such that F(x) =
_
x

## p(x)dx, the Stieltjes integral reduces to a Riemann integral

_
f(x)dF(x) =
_
f(x)p(x)dx.
Theorem 17 Let be a step function dened on [a, b] with jumps
k
at x
k
. Then
_
b
a
f(x)d(x) =

n
k=1
f(x
k
)
k
.
19
4 Sample Density Functions
Recall that F

## = f. This suggests that we can use the derivative of F

n
to estimate f. In
particular, we consider
f
n
(x) =
F
n
(x +b
n
) F
n
(x b
n
)
2b
n
.
Observe that 2nb
n
f
n
(x) Bin(n, F(x +b
n
) F(x b
n
)) and we have
Ef
n
(x) =
1
2nb
n
n[F(x +b
n
) F(x b
n
)]

1
2nb
n
n 2b
n
f(x) if b
n
0,
var(f
n
(x)) =
1
(2nb
n
)
2
n[F(x +b
n
) F(x b
n
)] [1 F(x +b
n
) +F(x b
n
)]

1
4nb
2
n
2b
n
f(x) =
f(x)
2nb
n
if nb
n
.
A natural question to ask is to nd an optimal choice of b
n
. To do so, we need to nd the
magnitude of bias, Ef
n
(x) f(x).
Since the above estimate can be viewed as the widely used kernel estimate with kernel
W(z) = 1/2 if |z| 1 and = 0 otherwise, we will nd the the magnitude of bias for the
following kernel estimate of f(x) instead.
f
n
(x) =
1
nb
n
n

i=1
W
_
x X
i
b
n
_
,
where W is an integrable nonnegative weight function. Typically, W are chosen to be a
density function,
_
tW(t)dt = 0, and
_
t
2
W(t)dt = = 0. We have
Ef
n
(x
0
) =
1
b
n
EW
_
x
0
X
b
n
_
=
1
b
n
_
W
_
x
0
x
b
n
_
f(x)dx =
_
W(t)f(x
0
b
n
t)dt
=
_
W(t)
_
f(x
0
) b
n
tf

(x
0

t
b
n
t)
_
dt
= f(x
0
) b
n
_
tW(t)f

(x
0

t
b
n
t)dt.
When
_
tW(t)dt = 0, we have
Ef
n
(x
0
) = f(x
0
) b
n
f

(x
0
)
_
tW(t)dt +o(b
n
).
When
_
tW(t)dt = 0 and
_
t
2
W(t)dt = 0, we have
Ef
n
(x
0
) =
_
W(t)
_
f(x
0
) b
n
tf

(x
0
) +
b
2
n
2
t
2
f

(x
0

t
b
n
t)
_
dt
= f(x
0
) +
b
2
n
2
f

(x
0
)
_
t
2
W(t)dt +o(b
2
n
).
Therefore, b
n
= O(n
1/3
) when
_
tW(t)dt = 0 (i.e., Assume that f

exists.), and b
n
=
O(n
1/5
) when
_
tW(t)dt = 0,
_
t
2
W(t)dt = 0 (i.e., Assume that f

exists.)
20
5 Applications of Order Statistics
The following is a brief list of settings in which order stastistics might have a signicant role.
1. Robust Location Estimates. Suppose that n independent measurements are
available, and we wish to estimate their assumed common mean. It has long
been recognized that the sample mean, though attractive from many viewpoints,
suers from an extreme sensitivity to outliers and model violations. Estimates
based on the median or the average of central order statistics are less sensitive to
model assumptions. A particularly well-known application of this observation is
the accepted practice of using trimmed means (ignoring highest and lowest scores)
in evaluating Olympic gure skating performances.
2. Detection of Outliers. If one is confronted with a set of measurements and is
concerned with determining whether some have been incorrectly made or reported,
attention naturally focuses on certain order statistics of the sample. Usually the
largest one or two and/or the smallest one or two are deemed most likely to be
outliers. Typically we ask questions like the following: If the observations really
were iid, what is the probability that the latgest order statistic would be as large
as the suspiciously large value we have observed?
3. Censored Sampling. Considser life-testing experiments, in which a xed number
n of items are placed on test and the experiment is terminated as soon as a
prescribed number r have failed. The observed lifetimes are thus X
1:n

X
r:n
r+1:n
X
n:n
remain unobserved.
4. Waiting for the Big One. Disastrous oods and destructive earthquakes re-
cur throughout history. Dam construction has long focused on so called 100-year
ood. Presumably the dams are built big enough and strong enough to handle
any water ow to be encountered except for a level expected to occur only once
every 100 years. Architects in California are particularly concerned with construc-
tion designed to withstand the big one, presumably an earthquake of enormous
strength, perhaps a 100-year quake. Whether one agrees or not with the 100-
year diaster philosophy, it is obvious that designers of dams and skycrapers, and
even doghouses, should be concerned with the distribution of large order statistics
from a possibly dependent, possibly not identically distributed sequence.
After the disastrous ood of February 1st, 1953, in which the sea-dikes broke in
several parts of the Netherlands and nearly two thousand people were killed, the
Dutch government appointed a committee (so-called Delta-committee) to recom-
mend on an appropriate level for the dikes (called Delta-level since) since no specic
21
statistical study had been done to x a safer level for the sea-dikes before 1953.
The Dutch government set as the standard for the sea-dikes that at any time in a
given year the sea level exceeds the level of the dikes with probability 1/10, 000. A
statistical group from the Mathematical Centre in Amsterdam headed by D. van
Dantzig showed that high tides occurring during certain dangerous windstorms (to
ensure independence) within the dangerous wintermonths December, January and
February (for homogenity) follow closely an exponential distribution if the smaller
high tides are neglected.
If we model the annual maximum ood by a random variable Z, the Dutch gov-
ernment wanted to determine therefore the (1 q)-quantile
F
1
(1 q) = inf{t R : F(t) 1 q}
of Z, where F denotes the distribution of Z and q has the value 10
4
.
5. Strength of Materials. The adage that a chain is no longer than its weakest link
underlies much of the theory of strength of materials, whether they are threads,
sheets, or blocks. By considering failure potential in innitesimally small sections
of the material, on quickly is led to strength distributions associated with limits
of distributions of sample minima. Of course, if we stick to the nite chain with
n links, its strength would be the minimum of the strengths of its n component
6. Reliability. The example of a cord composed of n threads can be extended to
lead us to reliability applications of order statistics. It may be that failiure of one
thread will cause the cord to break (the weakest link), but more likely the cord
will function as long as k (a number less than n) of the threads remains unbroken.
as such it is an example of a k out of n system commonly discussed in reliability
settings. With regard to tire failure, the automobile is often an example of a 4
out of 5 system (remember the spare). Borrowing on terminology from electrical
systems, the n out of n system is also known as a series system. Any component
failure is disastrous. The 1 out of n system is known as a parallel system; it will
function as long as any of the components survives. The life of the k out of n
system is clearly X
nk+1:n
, the (n k + 1)st largest of the component lifetimes,
or, equivalently, the time until less than k components are functioning. But, in
fact, they can be regarded as perhaps complicated hierarchies of parallel and series
subsystems, and the study of system lifetime will necessarily involve distributions
of order statistics.
7. Quality Control. Take a comfortable chair and watch the daily production of
Snickers candy bars pass by on the conveyor belt. Each candy bar should weigh
22
2.1 ounces; just a smidgen over the weight stated on the wrapper. No matter
how well the candy pouring machine was just adjusted at the beginning of the
shift, minor uctuations will occur, and potentially major aberrations might be
encountered (if a peanut gets stuck in the control valve). We must be alert for
correctable malfunctions causing unreasonable variation in the candy bar weight.
Enter the quality control man with his

X and R charts or his median and R
charts. A sample of candy bars is weighed every hour, and close attention is paid
to the order statistics of the weights so obtained. If the median (or perhaps the
mean) is far from the target value, we must shut down the line. Either we are
turning out skinny bars and will hear from disgruntled six-year-olds, or we are
turning out overweight bars and wasting money (so we will hear from disgruntled
management). Attention is also focused on the sample range, the largest minus
the smallest weight. If it is too large, the process is out of control, and the widely
uctuating candy bar weights will probably cause problems further down the line.
So again we stop and seek to identify a correctable cause before restarting the
Snickers line.
8. Selecting the Best. Field trials of corn varieties involved carefully balanced ex-
periments to determine which of several varieties is most productive. Obviously we
are concerned with the maximum of a set of probability not identically distributed
variables in such a setting. The situation is not unlike the one discussed earlier in
the context of identication of outliers. In the present situation, the outlier (the
best variety) is, however, good and merits retention (rather than being discarded
or discounted as would be the case in the usual outliers setting). Another instance
in biology in which order statistics play a clear role involves selective breeding by
culling. Here perhaps the best 10% with respect to meatiness of the animals in
each generation are raised for breeding purposes.
Geneticists and breeders measure the eectiveness of a selection program by com-
paring the average of the selected group with the population average. This dier-
ence, expressed in standard deviation units, is known as the selection dierential.
Usually, the selected group consists of top or bottom order statistics. Without
loss of generality let us assume the top k order statistics are selected. Then the
selection dierential is
D
k,n
(, ) =
1

__
n

i=nk+1
X
i:n
_
1
k

_
,
where and are the population mean and standard deviation, respectively.
Breeders quite often use E(D
k,n
(, )) or D
k,n
(, ) as a measure of improvement
due to selection. If k = n [np], then except for a change of location and scale,
23
D
k,n
(, ) is a trimmed mean with p
1
= p and p
2
= 1 where p
1
and 1p
2
represent
the proportion of the sample trimmed at either ends.
9. Inequality of Measurement. The income distribution in Bolivia (where a few
individuals earn most of the money) is clearly more unequal than that of Swe-
den (where progressive taxation has a leveling eect). How does one make such
statements precise? The usual approach involves order statistics of the corre-
sponding income distributions. The particular device used is call a Lorenz curve.
It summarizes the percent of total income accruing to the poorest p percent of the
population for various values of p. Mathematically this is just the scaled integral
of the empirical quantile function, a function with jump X
(i:n)
at the point i/n;
i = 1, 2, . . . , n (where n is the number of individual incomes in the population). A
high degree of convexity in the Lorenz curve signals a high degree of inequality in
the income distribution.
10. Olympic Records. Bob Beamons 1968 long jump remains on the Olympic record
book. Few other records last that long. If the best performances in each Olympic
Games were modeled as independent identically distributed random variables, then
records would become more and more scarce as time went by. Such is not the case.
The simplest explanation involves improving and increasing populations Thus the
1964 high jumping champion was the best of, say, N
1
active international-caliber
jumpers. In 1968 there were more high-caliber jumpers of probably higher caliber
So we are looking, most likely, at a sequence of not identically distributed random
variables. But in any case we are focusing on maximum.
11. Alocation of Prize Money in Tournaments.
12. Characterizations and Goodness of Fit. The exponential distribution is fa-
mous for its so-called lack of memory. The usual model involves a light bulb or
other electronic device. The argument goes that a light bulb that has been in
service 20 hours is no more and no less likely to fail in the next minute than one
that has been in service for, say, 5 hours, or even, for that metter, than a brand
new bulb. Such a curious distributional situation is reected by the order statistics
from exponential samples. For example, if X
1
, . . . , X
n
are iid exponential, then
their spacings (X
(i)
X
(i1)
) are again exponential and, remarkably, are indepen-
dent. It is only in the case of exponential random variables that such spacings
properties are encountered. A vast literature of exponential characterizations and
related goodness-of-t tests has consequently developed. We remark in passing
that most tests of goodness of t for any parent distribution implicitly involve or-
der statistics, since they often focus on deviations between the empirical quantile
24
function and the hypothesized quantile function.
25
6 Extreme Value Theory
At the beginning of this Chapter, we describe some asymptotic results concerning the pth
sample quantile X
(r)
, where as n , r = [np] and 0 < p < 1. This is the so-called central
order statistic. When F is absolutely continuous with nite positive pdf at F
1
(p), X
(r)
is
asymptotically normal after suitable normalization.
When either r or nr is xed and the sample size n , X
(r)
is called extreme order
statistic. We treat the extreme case in Section 6.1. One important message about extreme
order statistics is that if the limit distribution exists, it is nonnormal and depends on F only
through its tail behavior.
When both r or n r approach innity but r/n 0 or 1, X
(r)
is called intermediate
order statistic. The limiting distribution of X
(r)
depends on the rate of growth of r, and it
could be either normal or nonnormal. Refer to Falk (1989) for further details.
6.1 Possible Limiting Distributions for the Sample Maximum
Now we discuss the possible nondegenerate limit distributions for X
(n)
. Denote the upper
limit of the support of F by F
1
(1). Suppose that F
1
(1) < . Observe that F
X
(n)
(x) =
[F(x)]
n
. Then X
(n)
P
F
1
(1) can be established easily. Recall an elementary result in
Calculus:
_
1
c
n
n
_
n
exp(c) if and only if lim
n
c
n
= c,
where {c
n
} be a sequence of real numbers. Consider the normalization [X
(n)
F
1
(1)]/b
n
and
n
1
c
n
= 1F(F
1
(1)+b
n
x). Depending on the tail of F, it is expected that dierent norming
constants b
n
and asymptotic distribution emerge. When F
1
(1) = , it is not clear how to
nd the limiting distribution of X
(n)
in general. In order to hope for a nondegenerate limit
distribution, we will have to appropriately normalize or standardize X
(n)
. In other words,
we look at the sequence {(X
(n)
a
n
)/b
n
, n 1} where a
n
represents a shift in location and
b
n
> 0 represents a change in scale. The cdf of the normalized X
(n)
is F
n
(a
n
+b
n
x). We will
(i) Is it possible to nd a
n
and b
n
> 0 such that F
n
(a
n
+ b
n
x) G(x) at all continuity
points of a nondegenerate cdf G?
(ii) What kind of cdf G can appear as the limiting cdf?
(iii) How is G is related to F; that is, given F can we identify G?
(iv) What are appropriate choices for a
n
and b
n
in (i)?
In order to answer these questions precisely and facilitate the ensuing discussion, we introduce
two denitions
26
Denition (domain of maximal attraction). A cdf F (discrete or absolutely continu-
ous) is said to belong to the domain of maximal attraction of a nondegenerate cdf G if there
exist sequence {a
n
} and {b
n
> 0} such that
lim
n
F
n
(a
n
+b
n
x) = G(x)
at all continuity points of G(x). If the above holds, we will write F D(G).
Denition Two cdfs F
1
and F
2
are said to be of the same type if there exist constants
a
0
and b
0
> 0 such that F
1
(a
0
+b
0
x) = F
2
(x).
If the random variable (X
(n)
a
n
)/b
n
has a limit distribution for some choice of
constants {a
n
}, {b
n
}, then the limit distribution must be of the form G
1
, G
2
, or G
3
, where
G
1
(x; ) =
_
_
_
0, x 0,
exp(x

) , x > 0,
G
2
(x; ) =
_
_
_
exp((x)

) , x < 0,
1, x 0,
and
G
3
(x) = exp
_
e
x
_
, < x < .
(In G
1
and G
2
, is a positive constant.) This result was established by Gnedenko (1943),
following less rigorous treatments by earlier authors.
Note that the above three families of distributions may be related to the exponential
distribution as follows. If Y follows an exponential distribution, then G
1
(x, ) is the distri-
bution function of Y
1/
, G21(x, ) is the distribution function of Y
1/
, and G
3
(x) is the
distribution function of log(Y ).
6.2 Asymptotic Theory
As a motivated example, consider that X
1
, . . . , X
n
are uniformly distributed over [0, 1]. Be-
cause the uniform distribution has an upper terminal 1, it follows easily that X
(n)
P
1. Now
we would like to know how fast X
(n)
tends to 1. Alternatively, we attempt to choose appro-
priate constants {b
n
} such that Z
n
= (1 X
(n)
)/b
n
will have a nondegenerate distribution.
Note that
P(Z
n
z) =
_
_
_
0 if z < 0
1 (1 b
n
z)
n
if b
1
n
> z > 0
.
Choose b
n
= n
1
and then
(1 z/n)
n
exp(z).
This concludes that n(1 X
(n)
) has a limiting exponential distribution with unit mean.
27
We now consider the case that X has an upper terminal and 1 F(x) a( x)
c
for some c > 0 as x . Again, consider Z
n
= ( X
(n)
)/b
n
where b
n
is to be chosen. We
have the cumulative distribution function
1 [F( b
n
z)]
n
1 [1 a(b
n
z)
c
]
n
for z 0 and b
n
z = O(1). We take ab
c
n
= n
1
to show that (X
(n)
)(na)
1/c
has the limiting
p.d.f. cz
c1
exp(z
c
).
Next, suppose that X does not have an upper terminal and that as x
1 F(x) ax
c
for some c > 0.
Note that F is a Pareto() cdf when 1 F(x) = x

## , x 1, > 0, where is the shape

parameter. Consider Z
n
= X
(n)
/b
n
, we have that
Pr(Z
n
z) = [1 {1 F(b
n
z)}]
n
{1 a(b
n
z)
c
}
n
.
The choice b
n
= (an)
1/c
then gives the limiting result
Pr(Z
n
z) exp(z
c
).
This covers as a special case the Cauchy distribution with c = 1.
Finally, suppose that 1 F(x) tends to zero exponentially fast as x . We return
to the more general standardization Z
n
= (X
(n)
a
n
)/b
n
, when
Pr(Z
n
z) = (1 exp[log{1 F(a
n
+b
n
z)}])
n
. (5)
The crucial values of X
(n)
are those close to F
1
(1 1/n), which we take to be a
n
. Observe
that
1 F(a
n
+b
n
z) 1 {F(a
n
) +b
n
zf(a
n
))} = n
1
(1 nb
n
zf(a
n
))
and nb
n
zf(a
n
) 0. Then, expanding (??) in a Taylor series, we obtain
Pr(Z
n
z)
_
1 n
1
exp{b
n
znf(a
n
)}

n
,
so that with b
1
n
= nf(a
n
), we have
Pr(Z
n
z) exp(e
z
),
the corresponding p.d.f. being exp(z e
z
).
We now use two examples to show how to nd a
n
and b
n
.
Example 1 (Beta Distribution) Let F be a Beta(, ) cdf with density
f(x) = cx
1
(1 x)
1
I(0 < x < 1),
28
where c = ( +)/(() + ()). As x 1

, f(x) c(1 x)
1
, and
1 F(x) c
_
1
x
(1 u)
1
du = c(1 x)

/.
Find appropriate choices of a
n
and b
n
.
Sol. Note that a
n
= 1 and 1 F(1 b
n
) = n
1

n
/(nc), so we may take
b
n
=
_
()( + 1)
n( +)
_
1/
.
For UNIF(0, 1), = = 1. Hence b
n
= n.
Example 2 (Weibull Distribution) Let F be a Weibull() cdf where is the shape
parameter, that is F(x) = 1 exp(x

## ) for x > 0 and > 0. Find appropriate choices of

a
n
and b
n
.
Sol. Note that F(a
n
) = 1 n
1
n
= (log n)
1/
. Hence, b
1
n
= nf(a
n
) =
(log n)
(1)/
.
For exponential distribution, = 1 and hence a
n
= log n and b
n
= 1.
Example 3 (Standard Normal Distribution) Let F be a standard normal distribution.
Find appropriate choices of a
n
and b
n
.
Sol. Using LHospitals rule, we obtain
1 F(x)
1
x
f(x)
when x is large. Note that F(a
n
) = 1n
1
n
=

2 log n.
Hence, b
1
n
= nf(a
n
) = 1/

## 2 log n. Refer to p.99 of Ferguson (1996) for the detail of calcu-

lation.
As we know, Cauchy distribution is a special case of t-distribution. For t
v
distribution,
it has density
f(x) =
c
(v +x
2
)
(v+1)/2
cx
(v+1)
.
Can you gure out a
n
and b
n
? Refer to p.95 of Ferguson (1996) for the detail of calculation.
7 Linear Estimation of Location
Let X
1
, X
2
, . . . , X
n
be a random sample from the absolutely continuous cdf F(y; ) where is
the location parameter. For several distributions, linear functions of order statistics provide
good estimator of . Suppose a
in
s form a (double) sequence of constants. The statistic
L
n
=
n

i=1
a
in
X
(i:n)
is called an L statistic. When used as an estimator, it is often referred to as an L estimator.
A wide variety of limit distributions are possible for L
n
. For example, when a
in
is zero
for all but one i, 1 i n, L
n
is a function of a single order statistic. We have seen in
29
previous section the possible limit distributions for X
(i:n)
which depend on how i is related
to n. If L
n
is a function of a nite number of central order statistics, the limit distribution
is normal, under mild conditions on F.
Even when the a
in
are nonzero for many is, L
n
turns out to be asymptotically normal
when the weights are reasonably smooth. In order to make this requirement more precise, let
us suppose a
in
is of the form J(i/(n +1))/n, where J(u), 0 u 1, is the associated weight
function. In other words, we assume now that L
n
can be expressed as
L
n
=
1
n
n

i=1
J
_
i
n + 1
_
X
(i:n)
.
The asymptotic normality of L
n
has been established either by putting conditions on the
weights or the weight function.
Suppose that and the population median (= F
1
(1/2)) coincide. Let us assume
that the variance
2
is nite and f() is nite and positive. For simplicity, let us take the
sample size n to be odd. While

X
n
is an uubiased, asymptotically normal estimator of with
variance V ar(

X
n
) =
2
/n,

X
n
is asymptotically unbiased and normal. If the population pdf
is symmetric (around ),

X
n
is also unbiased. Further, V ar(

X
n
) {4n[f()]
2
}
1
. Thus, as
an estimator , the sample median would be more ecient than the sample mean, at least
asymptotically, whenever [2f()]
1
< . This condition is satised, for example, for the
Laplace distribution with pdf f(x; ) =
1
2
exp(|x|), < x < . For this distribution,
we know that

X
n
is the maximum-likelihood estimator of , and that it is robust against
outliers. Further, since f() = 1/2, we can construct condence intervals for using the fact
that

n(

X
n
) is asymptotically standard normal.
30
References
[1] Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N. (1992). A First Course in Order
Statistics. John Wiley & Sons, Inc.
[2] Chung, K.L. (1974). A Course in Probability Theory. 2nd ed., Academic Press, New
York.
[3] Cramer, H. and Wold, H. (1936). Some theorems on distribution functions. J. London
Math. Soc. 11 290-295.
[4] Dvoretzky, A., Kiefer, J., and Wolfowitz, J. (1956). Asymptotic minimax character of
the sample distribution function and of the classical multinomial estimator. Ann. Math.
Statist. 27 642-669.
[5] Falk, M. (1989). A note on uniform asymptotic normality of intermediate order statistics.
Ann. Inst. Statist. Math. 41 19-29.
[6] Ferguson, T.S. (1996). A Course in Large Sample Theory. Chapman & Hall.
[7] Gnedenko, B.V. (1943). Sue la distribution limite du terme maximum dun serie aleatoire,
Ann. Math. 44 423-453.
[8] Wilk, M.B. and Gnanadesikan, R. (1968). Probability plotting methods for the analysis
of data. Biometrika 55 1-17.
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