hw6 - T ) = p (1-p ). (c) Show that P ( T = 1 | W = w ) = w...

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(i) textbook, chapter 8, #16(d), also show that the pdfs form an exponential family and find a sufficient and complete statistic. #69. #71, also show the pdfs form an exponential family and find a sufficient and complete statistic. #72, also show that the gamma distribution form a 2-parameter exponential family and show that Q n i =1 X i and n i =1 X i are sufficient and complete. (ii) Let X 1 ,...,X n be an i.i.d. sample from N ( θ, 1). (a) Show that X 2 - 1 n is a UMVUE of g ( θ ) = θ 2 . (b) Identify whether the Cramer-Rao bound is attained for the UMVUE. (iii) Let Y 1 ,...,Y n be an i.i.d. sample from Bernoulli distribution B ( p ). Find a UMVUE of p (1 - p ), which is a term in the variance of Y i or W = n i =1 Y i , by the following steps: (a) Show that W is a sufficient and complete statistic. (b) Let T = ( 1 , if Y 1 =1 and Y 2 =0 0 , otherwise Show that E(
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Unformatted text preview: T ) = p (1-p ). (c) Show that P ( T = 1 | W = w ) = w ( n-w ) n ( n-1) . (d) Show that E( T | W ) = n n-1 W n 1-W n = n n-1 Y (1-Y ) and, explain why n Y (1-Y ) ( n-1) is a UMVUE of p (1-p ). (iv) Let Y 1 ,...,Y n be an i.i.d. sample from the pdf: f ( y | ) = ( 3 y 2 3 , if 0 y , otherwise where 0 < < . (a) Check whether the pdfs form an exponential family. (b) Show that Y ( n ) = max { Y 1 ,...,Y n } is a sucient statistic. (c) Show that Y ( n ) has pdf: f Y ( n ) ( y | ) = ( 3 ny 3 n-1 3 n , if 0 y , otherwise (d) Show that Y ( n ) is complete by denition. (e) Find a UMVUE of ....
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