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