# hw6 - T = p(1-p(c Show that P T = 1 | W = w = w n-w n n-1(d...

This preview shows page 1. Sign up to view the full content.

(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(
This is the end of the preview. Sign up to access the rest of the document.

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 | θ ) = ( 3 y 2 θ 3 , if 0 ≤ y ≤ θ , otherwise where 0 < θ < ∞ . (a) Check whether the pdfs form an exponential family. (b) Show that Y ( n ) = max { Y 1 ,...,Y n } is a suﬃcient statistic. (c) Show that Y ( n ) has pdf: f Y ( n ) ( y | θ ) = ( 3 ny 3 n-1 θ 3 n , if 0 ≤ y ≤ θ , otherwise (d) Show that Y ( n ) is complete by deﬁnition. (e) Find a UMVUE of θ ....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online