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stat210a_fall07_hw6

# stat210a_fall07_hw6 - UC Berkeley Department of Statistics...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 6 Fall 2007 Issued: Thursday, October 4 Due: Thursday, October 11 Reading: Keener, Chapter 11. Bickel and Doksum; § 2.1–2.3. § 3.1–3.3 Problem 6.1 Consider n i.i.d. samples X Uni[ θ - 1 2 , θ + 1 2 ], where θ Ω = R . Show that δ ( X 1 , . . . , X n ) = 1 2 [ X (1) + X ( n ) ] is the unique minimax estimator under squared error loss. Problem 6.2 (Recall that an equalizer rule δ has R ( θ, δ ) = c for all θ Ω.) Consider the decision- theoretic problem with parameter space Ω = [0 , 1), action space A = [0 , 1] and loss function L ( θ, a ) = ( θ - a ) 2 / (1 - θ ). Suppose that X has the distribution p ( x ; θ ) = (1 - θ ) θ x , x = 0 , 1 , 2 , . . . (1) (a) Write the risk function R ( θ, δ ) as a power series in θ . (b) Show that the only non-randomized equalizer rule is given by δ (0) = 1 2 , and δ ( i ) = 1 for i = 1 , 2 , . . . . (c) Show that a rule δ is Bayes w.r.t. a distribution Λ if and only if δ (0) = μ 1 and δ ( i ) = μ i i - 1 for i = 1 , 2 , 3 . . . , where μ i = E Λ i ] are the (power) moments of Λ.

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stat210a_fall07_hw6 - UC Berkeley Department of Statistics...

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