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Unformatted text preview: 1. Problem 7.2.8 in the text. Solution (a) The bias of the estimate is bias = E (ˆ p ) p = E ( X/ 11) p = 10 11 p p = p/ 11 (b) The variance of the estimate is Var(ˆ p ) = Var( X/ 11) = 1 121 Var( X ) = 10 p (1 p ) 121 (c) The mean square error is MSE = E (ˆ p p ) 2 = Var(ˆ p ) + bias 2 = 10 p (1 p ) 121 + p 2 121 = 10 p 9 p 2 121 (d) The bias of the estimator ˆ p 1 = X/ 10 is E (ˆ p 1 ) p = E ( X/ 10) p = 0 The mean square error is therefore MSE(ˆ p 1 ) = Var(ˆ p 1 ) + bias 2 = 1 100 Var( X ) = p (1 p ) 10 The di erence between mean square errors is given by 10 p 9 p 2 121  {z } MSE for ˆ p p (1 p ) 10  {z } MSE for ˆ p 1 = 31 p 2 21 p 1210 we want to know when the quantity 31 p 2 21 p 1210 is negative, which only happens when 31 p 2 21 p is negative; in other words, 31 p 2 21 p ≤ p (31 p 21) ≤ 31 p 21 ≤ p ≤ 21 / 31 2. Suppose that a 6sided die is rolled n times, showing values X 1 ,...,X n . Make a point estimate of the probability of rolling a 1 on the die, and write the standard error of this estimate. 1 Solution The number of 1 's shown on the die after n rolls is a B ( n,p = 1 / 6) random variable, which we can denote by Y . From lecture 11, slides 79, our estimate is the sample proportion ˆ p = Y/n , which has the approximate distribution ˆ p ∼ N p, p (1 p ) n = N 1 / 6 , 5 36 n 3. Determine the maximum likelihood estimator of θ when X 1 ,...,X n is a random variable with density function f ( x ) = 1 2 e x θ  for∞ < x < ∞ . Solution Write the likelihood function L ( x 1 ,...,x n ; θ ) = 1 2 e x 1 θ  ··· 1 2 e x n θ  = 2 n e (  x 1 θ  + ··· +  x n θ  ) The loglikelihood function is ‘ ( x 1 ,...,x n ; θ ) = log 2 n e (  x 1 θ  + ··· +  x n θ  ) = n log 2  x 1 θ   ···   x n θ  We want to maximize this function, which is a bit sticky because of the absolute values. Note that d dθ  x i θ  = ( 1 if θ < x i 1 if θ > x i and therefore d‘ dθ = ( #entries where θ > x i ) ( #entries where θ < x i ) To maximize this function we set...
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This note was uploaded on 02/19/2011 for the course IE 4521 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
 Staff

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