stat210a_fall07_hw9 - UC Berkeley Department of Statistics...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 9 Fall 2007 Issued: Thursday, November 1 Due: Thursday, November 8 Reading: § 3.5, § 4.4 Problem 9.1 Suppose that for each θ 0 Θ, the set A ( θ 0 ) is the acceptance region of a level α test for H 0 : θ = θ 0 . For each sample point x , define S ( x ) = { θ Θ | x A ( θ ) } . Show that the random set S ( X ) is a confidence set of level 1 - α . Problem 9.2 Let X 1 ,...,X n be an i.i.d. sample from the uniform distribution on [0 ]. (a) Consider the problem of testing H 0 : θ θ 0 versus H 1 : θ > θ 0 . Show that any test δ for which E θ 0 [ δ ( X )] = α , E θ [ δ ( X )] α for all θ θ 0 and δ ( x ) = 1 when x ( n ) = max { x 1 ,...,x n } > θ 0 is UMP at level α . (b) Now consider the problem of testing H 0 : θ = θ 0 against H 1 : θ 6 = θ 0 . Show that a unique UMP test exists, and is given by δ ( x ) = ( 1 if x ( n ) > θ 0 or x ( n ) < θ
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This note was uploaded on 10/17/2009 for the course STAT 210a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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stat210a_fall07_hw9 - UC Berkeley Department of Statistics...

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