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Unformatted text preview: Second Edition 73 The likelihood function is L (  x ) = n Y i =1 1 I [0 , ] ( x i ) = 1 n I [0 , ] ( x ( n ) ) I [0 , ) ( x (1) ) , where x (1) and x ( n ) are the smallest and largest order statistics. For x ( n ) , L = 1 / n , a decreasing function. So for x ( n ) , L is maximized at = x ( n ) . L = 0 for < x ( n ) . So the overall maximum, the MLE, is = X ( n ) . The pdf of = X ( n ) is nx n 1 / n , 0 x . This can be used to calculate E = n n + 1 , E 2 = n n + 2 2 and Var = n 2 ( n + 2)( n + 1) 2 . is an unbiased estimator of ; is a biased estimator. If n is large, the bias is not large because n/ ( n + 1) is close to one. But if n is small, the bias is quite large. On the other hand, Var < Var for all . So, if n is large, is probably preferable to ....
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.
 Spring '12
 Dr.Hackney
 Statistics

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