hwk1_soln - x ) = ( P ( X 1 x )) n = x b n , x b, taking...

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IEOR E4702 Statistical Inference for Financial Engineering HWK Solution 1 1. The total return of these two managers are: Manager A : p 1 60% + (1 p 1 ) 15% ; Manager B : p 2 35% + (1 p 2 ) 5% . When the Simpson’s paradox appears, p 1 60% + (1 p 1 ) 15% <p 2 35% + (1 p 2 ) 5% , which gives us 0 . 45 p 1 +0 . 10 < 0 . 3 p 2 , i.e. p 2 > 3 2 p 1 + 1 3 . Also, p 1 and p 2 must satisfy 0 p 1 ,p 2 1 . 2. From the de f nition of variance Var ( ˆ θ θ )= E [( ˆ θ θ ) 2 ] ( E [ ˆ θ θ ]) 2 . Therefore, rewrite the above equation we have MSE ( ˆ θ )= E [( ˆ θ θ ) 2 ]= Var ( ˆ θ θ )+( E [ ˆ θ θ ]) 2 = Var ( ˆ θ )+( Bias ( ˆ θ )) 2 . The last equality holds because θ is a constant. 3. (1) Let X (1) =m in( X 1 ,...,X n ) and X ( n ) =max( X 1 ,...,X n ) . The likelihood is given by L ( b )= n Y i =1 1 b I { 0 X i b } = 1 b n I © 0 X (1) X ( n ) b ª . Although the function L ( b ) is not di f erentiable, L ( b ) is a decreasing function of b if b X ( n ) , and L ( b )=0 if b<X ( n ) . Therefore, the maxima is obtained at b = X ( n ) ;i . e . th eMLEi s ˆ b =
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(2) Since P ( ˆ b x )= P ( X 1 x, X 2 x,. ..,X n
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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.

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hwk1_soln - x ) = ( P ( X 1 x )) n = x b n , x b, taking...

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