Theory of Statistics - George Mason University

1.3 Sequences of Events and of Random Variables 97
Properties of quantiles, of course, are different for discrete and continuous
distributions. In the following, for 0 < π < 1 we will assume that F(xπ)
is twice differentiable in some neighborhood of xπ and F ′′ is bounded and
F ′ (xπ) > 0 in that neighborhood. Denote F ′ (x) as f(x), and let Fn(x) =
#{X1, . . ., Xn}/n. Now, write the k th order statistic as
X(k:n) = xπ − Fn(xπ) − π
f(xπ)
+ Rn(π). (1.200)
This is called the Bahadur representation, after Bahadur (1966), who showed
that Rn(π) → 0 as n → ∞. Kiefer (1967) determined the exact order of Rn(π),
so equation (1.200) is sometimes called the Bahadur-Kiefer representation.
The Bahadur representation is useful in studying asymptotic properties of
central order statistics.
There is some indeterminacy in relating order statistics to quantiles. In
the Bahadur representation, for example, the details are slightly different if
nπ happens to be an integer. (The results are the same, however.) Consider
a slightly different formulation for a set of m order statistics. The following
result is due to Ghosh (1971).
Theorem 1.48
Let X1, . . ., Xn be iid random variables with PDF f. For k = n1, . . ., nm ≤ n,
let λk ∈]0, 1[ be such that nk = ⌈nλk⌉+1. Now suppose 0 < λ1 < · · · < λm < 1
and for each k, f(xλk) > 0. Then the asymptotic distribution of the random
m-vector
n 1/2 (X(n1:n) − xλ1), . . ., n 1/2
(X(nm:n) − xλm)
is m-variate normal with mean of 0, and covariance matrix whose i, j element
is
λi(1 − λj)
.
f(xλi)f(xλj)
For a proof of this theorem, see David and Nagaraja (2003).
A sequence of extreme order statistics {X(k:n)} is one such that k/n → 0 or
k/n → 1 as n → ∞. Sequences of extreme order statistics from a distribution
with bounded support generally converge to a degenerate distribution, while
those from a distribution with unbounded support do not have a meaningful
distribution unless the sequence is normalized in some way. We will consider
asymptotic distributions of extreme order statistics in Section 1.4.3.
We now consider some examples of sequences of order statistics. In Examples
1.27 and 1.28 below, we obtain degenerate distributions unless we
introduce a normalizing factor. In Example 1.29, it is necessary to introduce
a sequence of constant shifts.
Example 1.27 asymptotic distribution of min or max order statistics
from U(0, 1)
Suppose X1, . . ., Xn are iid U(0, 1). The CDFs of the min and max, X(1:n)
and X(n:n), are easy to work out. For x ∈ [0, 1],
Theory of Statistics c○2000–2013 James E. Gentle