Lecture 19 - Order Statistics

Lecture 19 - Order Statistics - 1 ORDER STATISTICS Lecture...

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Lecture 19 ORIE3500/5500 Summer2011 Li 1 Order Statistics Here we will learn how to formally treat the order statistics of the generally distributed random variables. Suppose X 1 ,X 2 ,...,X n are i.i.d. random variables with density function f X and cdf F X . The ordered random variables are denoted by X ( i ) ,i = 1 ,...,n which are just the random variables X i ’s ordered such that X (1) X (2) ≤ ··· ≤ X ( n ) . In particular X (1) = min( X 1 ,...,X n ) and X ( n ) = max( X 1 ,...,X n ). We have already seen before that F X (1) ( x ) = 1 - (1 - F X ( x )) n . and taking derivative we get f X (1) ( x ) = n (1 - F X ( x )) n - 1 f X ( x ) . Example If X 1 ,...,X n are i.i.d N (0 , 1) random variables , then the density of X (1) is given by f X (1) ( x ) = n 2 π 1 - Φ( x ) · n - 1 e - x 2 / 2 Similarly the cdf of the n th order statistic is F X ( n ) ( x ) = ( F X ( x )) n , and taking derivative we get f X ( n ) ( x ) = n ( F X ( x )) n - 1 f X ( x ) . We can ﬁnd the distribution of any ordered statistic. Let us ﬁrst compute the cdf of X ( k ) . Note that the event { X ( k ) x } is same as the event where at least k among X 1 ,...,X n is less than or equal to x . We can view this binomial trial: every observation has probability F X ( x

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Lecture 19 - Order Statistics - 1 ORDER STATISTICS Lecture...

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