University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou
Order statistics
Why order statistics?
We may be interested in
the fastest time in an automobile race,
the heaviest mouse among a group of mice fed on a certain diet,
the earliest time an electronic system fails,
the 1st or nth order statistics (could be estimates of parameters) etc.
Statistics 100B
Theory:
Let X1, X2, · · · , Xn denote independent continuous random variables with cdf
pdf f(x). We will denote the ordered random variables with X(1),X(2),···,X(n), where X(1) ≤ X(2) ≤ ··· ≤ X(n) or X(1) = min(X1,X2,···,Xn) and X(n) = max(X1,X2,···,Xn). We call X(1) the first order statistic and X(n) the nth order statistic. Similarly, X(j) is the jth order statistic. We want to find the pdf of X(1),X(n),X(j), but also joint pdf functions that involve order statistics.
Useful results (see class notes for proofs):
a. Probability density function of the 1st order statistic.
gX(1) (x) = n [1 − FX (x)]n−1 fX (x)
b. Probability density function of the nth order statistic.
gX(n) (x) = n [FX (x)]n−1 fX (x)
c. Probability density function of the jth order statistic.
gX (x) = n! [FX (x)]j−1 [1 − FX (x)]n−j fX (x) (j) (n − j)!(j − 1)!
d. Joint probability density function of X(1), X(2), · · · , X(n). gX(1),X(2),…,X(n) (x1, x2, . . . , xn) = n!fX (x1)fX (x2) . . . fX (xn)
e. Joint probability density function of X(i), X(j), with 1 ≤ i < j ≤ n.
gX(i),X(j)(u,v)= n! fX (u)fX (v)[FX (u)]i−1[FX (v)−FX (u)]j−1−i[1−FX (v)]n−j (i−1)!(j −1−i)!(n−j )!
F (x) and
1
Example 1:
Electronic components of a certain type have a length life (in hours) X, that follows the exponential distribution with probability density given by
100
f(x)= 1 e− 1 x, x>0.
100
a. Suppose that 2 such components operate independently and in series in a certain system (that is, the system fails when either component fails). Find the density function for the length of life of the system.
b. Suppose that 2 such components operate independently and in parallel in a certain system (that is, the system does not fail until both components fail). Find the density function for the length of life of the system.
Example 2:
Let X1, X2, . . . , Xn i.i.d. U(0, θ). Find the pdf of X(1), X(n), X(j), and the joint pdf of X(1), X(n).
2