order statistics

Let X1,,Xn be random variablesMathworldPlanetmath with realizations in . Given an outcome ω, order xi=Xi(ω) in non-decreasing order so that


Note that x(1)=min(x1,,xn) and x(n)=max(x1,,xn). Then each X(i), such that X(i)(ω)=x(i), is a random variable. StatisticsMathworldMathworldPlanetmath defined by X(1),,X(n) are called order statisticsMathworldPlanetmath of X1,,Xn. If all the orderings are strict, then X(1),,X(n) are the order statistics of X1,,Xn. Furthermore, each X(i) is called the ith order statistic of X1,,Xn.

Remark. If X1,,Xn are iid as X with probability density functionMathworldPlanetmath fX (assuming X is a continuous random variable), Let Z be the vector of the order statistics (X(1),,X(n)) (with strict orderings), then one can show that the joint probability density function f𝐙 of the order statistics is:


where 𝒛=(z1,,zn) and z1<<zn.

Title order statistics
Canonical name OrderStatistics
Date of creation 2013-03-22 14:33:30
Last modified on 2013-03-22 14:33:30
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 62G30