ORDER STATISTICS 1 ORDER STATISTICS Let X 1

ORDER STATISTICS 1

ORDER STATISTICS • Let X 1, X 2, …, Xn be a r. s. of size n from a distribution of continuous type having pdf f(x), a<x<b. Let X(1) be the smallest of Xi, X(2) be the second smallest of Xi, …, and X(n) be the largest of Xi. • X(i) is the i-th order statistic. 2

ORDER STATISTICS • It is often useful to consider ordered random sample. • Example: suppose a r. s. of five light bulbs is tested and the failure times are observed as (5, 11, 4, 100, 17). These will actually be observed in the order of (4, 5, 11, 17, 100). Interest might be on the kth smallest ordered observation, e. g. stop the experiment after kth failure. We might also be interested in joint distributions of two or more order statistics or functions of them (e. g. range=max – min) 3

ORDER STATISTICS • If X 1, X 2, …, Xn is a r. s. of size n from a population with continuous pdf f(x), then the joint pdf of the order statistics X(1), X(2), …, X(n) is Order statistics are not independent. The joint pdf of ordered sample is not same as the joint pdf of unordered sample. Future reference: For discrete distributions, we need to take ties into account (two X’s being equal). See, Casella and Berger, 1990, pg 231. 4

Example • Suppose that X 1, X 2, X 3 is a r. s. from a population with pdf f(x)=2 x for 0<x<1 Find the joint pdf of order statistics and the marginal pdf of the smallest order statistic. 5

ORDER STATISTICS • The Maximum Order Statistic: X(n) 6

ORDER STATISTICS • The Minimum Order Statistic: X(1) 7

ORDER STATISTICS • k-th Order Statistic y 1 y 2 … yk-1 yk yk+1 … P(X<yk) yn P(X>yk) y # of possible orderings n!/{(k 1)!1!(n k)!} f. X(yk) 8

Example • Same example but now using the previous formulas (without taking the integrals): Suppose that X 1, X 2, X 3 is a r. s. from a population with pdf f(x)=2 x for 0<x<1 Find the marginal pdf of the smallest order statistic. 9

Example • X~Uniform(0, 1). A r. s. of size n is taken. Find the p. d. f. of kth order statistic. • Solution: Let Yk be the kth order statistic. 10

ORDER STATISTICS • Joint p. d. f. of k-th and j-th Order Statistic (for k<j) k-1 items y 1 y 2 … yk-1 P(X<yk) j-k-1 items 1 item yk yk+1 … yj-1 yj yj+1 P(yk<X<yj) f. X(yk) n-j items 1 item yn # of possible orderings n!/{(k 1)!1!(j-k-1)!1!(n j)!} y P(X>yj) f. X(yj) 11