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# HW1 - STA131C Spring 2007 Prof W Polonik HW 1 due in class...

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STA131C, Spring 2007 Prof. W. Polonik HW # 1 due: April 11, 2007 in class 1. For α, β R let Y i = α + β x i + ǫ i , i = 1 , . . . , n where ǫ 1 , . . . , ǫ n i.i.d. N (0 , 1) and x 1 , . . . , x n are known constants which are not all the same. (a) (15 pts) Derive the MLE (ˆ α, ˆ β ) for ( α, β ) . (b) (10 pts) Derive the distribution of ˆ β. HINT: You might want to use the facts that n i =1 ( Y i x i ) - n ¯ Y ¯ x = n i =1 Y i ( x i - ¯ x ) and n i =1 x i ( x i - ¯ x ) = n i =1 ( x i - ¯ x ) 2 where ¯ x = 1 n n i =1 x i and ¯ Y = 1 n n i =1 Y i . 2. Let X 1 , . . . , X n be a random sample from f ( x | θ ) = θ x e - θ x 2 2 1 (0 , ) ( x ) , θ > 0 . (a) (10 pts) Verify that f ( x | θ ) is a pdf. (b) (10 pts) Find a sufficient statistic for θ . (c) (10 pts) Find the MLE for θ. (d) (5 pts) Is the MLE minimal sufficient? 3. Let X 1 , . . . , X n be a random sample from a Bernoulli( θ ) , θ (0 , 1) . Consider the testing problem H 0 : θ 1 / 2 versus H 1 : θ > 1 / 2 . Let δ
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