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stat210a_midterm_solutions

# stat210a_midterm_solutions - UC Berkeley Department of...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Midterm Examination—-Solutions Fall 2006 Problem 1.1 [18 points total] Suppose that X i , i = 1 , . . . , n are i.i.d. samples from the uniform Uni[0 , θ ] distribution. (a) Find a one-dimensional sufficient statistic for estimating θ . (b) Compute the maximum likelihood estimate θ MLE based on ( X 1 , . . . , X n ). Using an elementary argument, show that θ MLE p θ * as n + . (c) Consider the estimator of θ given by δ ( X ) = 2 n n i =1 X i . Is it unbiased? Is it admissible under squared error loss? Justify your answers. Now suppose that we view the parameter as a random variable Θ, and assume a Pareto prior density of the form λ ( θ ) = γβ γ θ - γ - 1 I [ β θ ] , for all θ > 0 , where β > 0 and γ > 2 are fixed. (d) Compute the prior mean of the random variable Θ. (e) Compute the posterior distribution of Θ conditioned on X = ( X 1 , . . . , X n ). (f) Compute the Bayes estimate of Θ under quadratic loss. Hint: New calculation may not be required given previous parts to the question. Solution 1.1: (a) By independence, we have p ( x ; θ ) = n i =1 1 θ I [ x i θ ] for x i 0 = θ - n I [max( x i ) θ ] , so that Z = max { X 1 , . . . , X n } is sufficient by the factorization criterion. (b) From part (a), the log likelihood takes the form L ( θ ) = - n log( θ ) for θ max { X i } , and -∞ otherwise, so that the MLE is given by θ MLE = max { X 1 , . . . , X n } . For any (0 , θ ), we compute P [ | max X i - θ | > ] = n i =1 P [ X i θ - ] = 1 - θ n 0 as n + , so that consistency of the MLE follows. 1

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(c) We compute E [ δ ( X )] = 2 n n i =1 E [ X i ] = 2 n ( n θ 2 ) = θ, so that the estimator is unbiased. However, since Z = max { X i } is sufficient from part (a) and this estimator depends on other information, the Rao-Blackwell theorem dic- tates that we can construct a better estimator δ ( X ) = E [ δ ( X ) | Z ]. The strict convexity of quadratic loss ensures that δ will dominate δ , so that δ must be inadmissible. (d) We compute the prior mean E [Θ] = + β θλ ( θ ) = γβ γ + β θ - γ = γβ γ - 1 .
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stat210a_midterm_solutions - UC Berkeley Department of...

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