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Unformatted text preview: x ) = ( P ( X 1 x )) n = x b n , x b, taking the f rst derivative yields the density of the MLE b n 1 b n x n 1 , x b. Thus, by using this density we have E h b i = Z b xn 1 b n x n 1 dx = n b n Z b x n dx = b n n + 1 6 = b. Therefore, b is a biased estimator. (3) The MLE b = X ( n ) is consistent. In fact, we can prove this by observing that for any < < b , no matter how small is, we always have P ( b b b ) = Z b b n 1 b n x n 1 dx = b n ( b ) n b n = 1 1 b n 1 , as the sample size n . Thus, we must have b b in probability. 2...
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This note was uploaded on 10/18/2010 for the course IEOR 4702 taught by Professor Kou during the Spring '10 term at Columbia.
 Spring '10
 kou
 Financial Engineering

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