advertisement

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