HW4 - (c Is it true that q ˆ θ Bayes = E q θ | x where...

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ORIE6700 Fall 2010 Homework 4 x ; just take this as given, don’t derive. Also, you may use the following fact. If Y i iid f ( x ) for some density function f , then the density of max i =1 ,...,n Y i has the density function nF ( x ) n - 1 f ( x ) where F is the CDF of f . 2. Let X 1 , . . . , X n be iid Bernoulli taking values 0,1 with probabilities 1 - θ, θ. Let the loss function be l ( θ, δ ) = ( θ - δ ) 2 . (a) (Do not hand in, done in class) Suppose the prior density π ( θ ) is the Beta( a, b ) density. Find the posterior density. (b) Using the same prior, what is the Bayes estimate of the Bernoulli variance q ( θ ) = θ (1 - θ ) . You will need the fact that, for any function g ( θ ), the Bayes estimate of g ( θ ) under quadratic loss is the posterior mean E ( g ( θ ) | x ) = Z g ( θ ) π ( θ | x )
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Unformatted text preview: (c) Is it true that q ( ˆ θ Bayes ) = E ( q ( θ ) | x ) where ˆ θ Bayes is the Bayes posterior mean estimate of θ ? 3. Suppose that X 1 , . . . , X n is a sample from a population with density f ( x | θ ) = θx θ-1 , < x < 1 , θ > 1 so that each X i is beta( θ, 1). Let T ( ~ X ) =-1 n n X i =1 log X i . (a) Show E ( T ( ~ X )) = 1 /θ , and Var( T ( ~ X )) = 1 /nθ 2 . (You should not need a lot of calculations here!) 4. C & B 10.3, p.505 5. C & B 2.33 a-c; read page 62. 6. Use the expression for the MGF of the natural suﬃcient statistic of an exponential family to compute the MGF of the suﬃcient statistic when doing iid sampling from (a) binomial (b) Poisson....
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This note was uploaded on 12/14/2010 for the course ORIE 6700 taught by Professor Woodard during the Fall '10 term at Cornell.

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