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
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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