Page 231 - Elements of Distribution Theory

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7.5 Order Statistics 217

Suppose that testing is continued until m bugs have been discovered. Then S 1 , the time
needed to find the first discovered bug, is the first order statistic Z (1) ; S 2 , the time needed
to find the first two bugs, is the second order statistic Z (2) , and so on. Thus, a statistical
analysis of this model would require the distribution of (Z (1) ,..., Z (m) ).

Example 7.21 (The sample range). Let X 1 ,..., X n denote independent, identically dis-
tributed random variables. One measure of the variability in the data {X 1 ,..., X n } is the
sample range, defined as the difference between the maximum and minimum values in the
sample; in terms of the order statistics, the sample range is given by X (n) − X (1) .

The distribution theory of the order statistics is straightforward, at least in principle. Let
F denote the distribution function of X j , j = 1,..., n. The event that X (n) ≤ t is equivalent
to the event that X j ≤ t, j = 1,..., n. Hence, X (n) has distribution function F (n) ,given by

n
F (n) (t) = F(t) .

Similarly, the event that X (n−1) ≤ t is equivalent to the event that at least n − 1ofthe X j
are less than or equal to t. Hence, X (n−1) has distribution function F (n−1) ,given by
n
F (n−1) (t) = F(t) + nF(t) n−1 (1 − F(t)).
This same approach can be used for any order statistic. The result is given in the following
theorem; the proof is left as an exercise.

Theorem 7.6. Let X 1 , X 2 ,..., X n denote independent, identically distributed real-valued
random variables, each with distribution function F. Then the distribution function of X (m)
is given by F (m) where

n
n i n−i
F (m) (t) = F(t) (1 − F(t)) , −∞ < t < ∞.
i
i=m
Example 7.22 (Pareto random variables). Let X 1 , X 2 ,..., X n denote independent, iden-
tically distributed random variables, each with an absolutely continuous distribution with
density function

θx −(θ+1) , x > 1,

where θ is a positive constant. Recall that this is a Pareto distribution; see Example 1.28.
The distribution function of this distribution is given by
t

F(t) = θx −(θ+1) dx = 1 − t −θ , t > 1.
1
Hence, the distribution function of X (m) , the mth order statistic, is given by

n
n −θ i −θ n−i
F (m) (t) = (1 − t ) (t ) , t > 1.
i
i=m