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EXAM I - STAT 543 February 15, 2006 Name: Be sure to show all work t o receive partial credit. Relax and GOOD LUCK! 1. Let X I , . . . , X, be a random sample from a GAMMA(a,9 ) distribution, 9 > 0, where a is a known constant. Assume that the Cram&-Rao regularity conditions are satisfied. (a) Find the Cramkr-Rao lower bound for estimating 9. - CRLB'& 0 \ 5 t~T,(wl - - - 3 - a ~ =o oa -- as -- a+ (bII k l - 3 -X (b) Find the UMVUE of 9. Justify your answer. T r y p)lr : & H = xfl 3 %=A LX Q9 2. Let X1,X 2 , .. . , be a sequence of iid random variables with common pdf f ( ~ 1 0= ) 0 > 0. Also let y(0) _= log(1 { ~ ~ if 00 < x -5 0 ~ otherwise, + 02). (a) Find the Method of Moments estimator T, of y(0) based on X I , . . . , X,. (b) Show that {T,) is consistent for y(0). Clearly state any standard results that you are using. P 6 s $ 0 RS iI*m N L L U , ?n* ~olj?) B~ (c) Show that the maximum likelihood estimator 6, of 0 based on X I , . . . , X, is given by 8, = max{X1, . . . , X,) . r The /ikd,h,~d-y41~hii , twk~bf (d) show that pB(en 5 y) = ify50 if 0 < y ify29 I0 (e) Using part(d) or otherwise, show that (9,) is consistent for 9. 3. Let X I , . . . , X, be a random sample from N(2,1/6), 6 > 0. We want to find the Bayes I ) , where a > 0 is a known constant. estimator of 6 w.r.t. the prior GAMMA(a + i, (a) Find the posterior distribution of 6. - - 02-1 e- p (b) Find the Bayes ,.+hPn 2 I q i a ,?= -I j f(~i-lt+~) '='-7 6)2/62. Justify your