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stat210a_fall07_hw9

# 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: Keener: Chapter 14; B & D: § 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 , θ ]. (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 0 ( α ) 1 /n 0 otherwise . Problem 9.3 Suppose that X 1 , . . . , X n are independent exponential random variables with E [ X i ] = βt i where t 1 , . . . , t n are known constants, and β > 0 is an unknown parameter.

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stat210a_fall07_hw9 - UC Berkeley Department of Statistics...

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