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# hw5 - E μ t | Y 1,Y t for t = 1 2 10 Attach your R code...

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STAT 428 Spring 2010 Homework 5 Mar 17 Homework 5 Due in class on Friday, Apr 2. 1. Normal Mean Shift Model : Let { Y 1 , Y 2 , . . . } be a sequence of independent N ( μ t , 1) random variables, where the μ t ’s undergo occasional changes. Y t = μ t + ε t , μ t = braceleftBigg μ t - 1 , with probability 0 . 9 , Z t , with probability 0 . 1 , where ε t are i.i.d. N (0 , 1), and Z t are i.i.d N (0 , 1). Suppose μ 0 = 0. Assume we observe Y 1 = 0 . 2, Y 2 = 0 . 1, Y 3 = 0 . 4, Y 4 = 1 . 3, Y 5 = 1 . 0, Y 6 = 1 . 2, Y 7 = 1 . 1, Y 8 = 0 . 1, Y 9 = 2 . 4, and Y 10 = 0 . 9. Use sequential importance sampling with sample size m = 50 to implement the filtering problem, i.e., estimating E ( μ
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Unformatted text preview: E ( μ t | Y 1 ,... ,Y t ), for t = 1 , 2 ,... , 10. Attach your R code and the results. 2. Suppose we want to estimate μ = E ( X ). Let C be a random variable. Assume we know E ( C ) = μ . Then we can form X ( b ) = bX + (1 − b ) C such that E [ X ( b )] = E ( X ). If C is correlated with X and V ar ( X ) n = Cov ( X,C ), show that we can always choose a proper b so that X ( b ) has smaller variance than X ....
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