stat210a_HW03 - UC Berkeley Department of Statistics STAT...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 3 - Solutions Fall 2009 Issued: Tuesday, September 15, 2009 Due: Tuesday, September 22, 2009 Problem 3.1 Let X N ( , 1). From, p ( x ) = 1 2 exp(- 1 2 x 2 + x- 2 2 ) , T ( X ) = X is sufficient by the factorization theorem. Notice that N ( , 1) is an exponential family with one parameter and one sufficient statistic and thus is of full rank, and T ( X ) = X is linearly independent. Hence T ( X ) = X is complete. From what is given, Z + - f ( x )exp( x )d x = 2 exp( 2 2 ) Z + - f ( x )exp( 1 2 x 2 ) 1 2 exp {- ( x- ) 2 2 } d x = 1 Since 1 2 R + - exp {- ( x- ) 2 2 } d x = 1, for any R , we have, Z + - [ f ( x )exp( 1 2 x 2 )- 1] 1 2 exp {- ( x- ) 2 2 } d x = 0 , for any R . From the completeness of T ( X ), we have, f ( x ) e 1 2 x 2- 1 = 0 , w.p.1 Therefore, f ( x ) = e- 1 2 x 2 . Problem 3.2 For > 0, let Z 1 , ,Z n U (0 , ), i.i.d. P ( Z 1 , ,Z n ) = n Y i =1 P ( Z i ) = n Y i =1 1 1 (0 < Z i < ) = 1 n 1 ( Z (1) > 0) 1 ( Z ( n ) < ) ....
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stat210a_HW03 - UC Berkeley Department of Statistics STAT...

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