midterm2old0

# midterm2old0 - ST 530 Name Midterm#2 All problem parts have...

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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 | θ ) = e - x θ θ (1 - e - 1 ) I (0 ) ( x ) , θ > 0 . (a) Does f ( x | θ ) 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 θ .

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2. Assume that X 1 , . . . , X n are i.i.d. Pareto(1 , β ), i.e., f ( x | β ) = β x β +1 I (1 , ) ( x ) . We assume that the parameter
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midterm2old0 - ST 530 Name Midterm#2 All problem parts have...

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