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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 11 Konrad Menzel March 17, 2009 Order Statistics Let X 1 ,...,X n be independent random variables with identical p.d.f.s f X 1 ( x ) = ... = f X n ( x ) - we’ll generally call such a sequence ”independent and identically distributed”, which is typically abbreviated as iid . We are interested in the function Y n = max { X 1 ,...,X n } i.e. Y n is the largest value in the sample. We can derive the c.d.f. of Y n using independence F Y n ( y ) = P ( Y n y ) = P ( X 1 y,X 2 y,...,X n y ) = P ( X 1 y ) P ( X 2 y ) ...P ( X n y ) = F X 1 ( y ) F X 2 ( y ) ...F X n ( y ) = [ F X ( y )] n Using the chain rule, we can obtain the p.d.f. of the maximum, d n 1 f Y n ( y ) = F Y n ( y ) = n [ F X ( y )] f X ( y ) dy Example 1 An old painting is sold at an auction. n identical bidders submit their bids B 1 ,...,B n independently, and the marginal c.d.f. of the bids is F B ( b ) . The potential buyer who submitted the highest bid gets to buy the painting and has to pay his bid (this type of auction is called Dutch, or first price auction). Then the revenue of the seller of the painting has p.d.f. is n 1 f Y ( y ) = f max { B 1 ,...,B n } ( y ) = n [ F B ( y )] f B ( y ) Now we can generalize this to other ranks in the sample, e.g. Y n 1 = ”The 2nd highest value in X 1 ,...,X n This random variable is called the ( n 1) th order statistic of X 1 ,...,X n , and we can state its p.d.f. Proposition 1 Let X 1 ,...,X n be an iid sequence of random variables with p.d.f. f X ( f ) and c.d.f. F X ( x ) . Then the k th order statistic Y k has p.d.f. f Y k ( y ) = k n [ F X ( y )] k 1 [1 F X ( y )] n k f X ( y ) k 1
Proof: We can split the experiment into two parts, (a) one of the X i ’s has to take the value y according to the density f X ( y ), and (b) given the value y , the other draws in the sequence have to be grouped around y in a way that makes y the k th smallest value in the sample. Part (b) is a binomial experiment in which the n trials correspond to the n draws of X 1 ,...,X n , and we define a ”success” in the i th round as the event ( X i y ). Since draws are independent and correspond to the same p.d.f. the parameter p in the binomial distribution is equal to F X ( y ). y being smaller or equal to the k th smallest value corresponds to at least k ”successes” in the binomial experiment, and therefore the corresponding c.d.f. is n F Y k ( y ) = n [ F X ( y )] l [1 F X ( y )] n l l l = k We can now obtain the p.d.f. by differentiating the c.d.f. with respect to y , using the product and the chain rule d f Y k ( y ) = F Y k ( y ) dy n n = n l [ F X ( y )] l 1 [1 F X ( y )] n l f X ( y ) n ( n l )[ F X ( y )] l [1 F X ( y )] n l 1 f X ( y ) = T 1 T 2 l l l = k l = k This expression looks complicated, but it turns out to be essentially a telescopic sum, so that most summands will drop out. Noting that for the second term, the summand corresponding to l = n is zero, we

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