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Unformatted text preview: ST 530 April 11, 2004 Name: Midterm #2 All problem parts have equal weight. In budgeting your time expect that some problems will take longer than others. You only need to answer 10 out of the 11 problem parts to get full credit. Justify all your claims! 1. Let X 1 , . . . , X n be i.i.d. random variables each with density f ( x | ) = e- x (1- e- 1 ) I (0 , ) ( x ) , > . (a) Does f ( x | ) constitute an exponential family? (b) Find a minimal sufficient statistics. (c) Is the minimal sufficient statistics you found complete? (Hint: It might help to show first that n i =1 X i /X ( n ) is ancillary.) (d) Find a MM (method of moments) estimator of . Solution: (a) No. The indicator I (0 , ) ( x ) depends on the parameter . (b) Notice that the joint density f ( x | ) = e- P n i =1 x i ( (1- e- 1 )) n I (0 , ) ( x ( n ) ) I (0 , ) ( x (1) ) . The ration f ( x | ) /f ( y | ) does not depend on if and only if n i =1 x i = n i =1 y...
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