All answers should be reasonably argued. Write the answer on the free
space following the problem. The 3 problems have same weight. All parts of
the same problem worth the same. You may use any thing written on paper
brought by you.
1. Let X1;:::;Xn be independent with Xi s ( i; ) (a gamma distribu-
tion with shaper parameter and scale parameter ), 1;:::; n are
known.
(a) Is this an exponential family of distributions?
(b) Is there a su cient statistics? A complete statistics?
(c) Find the MLE of .
(d) For which value of c, c=Pni=1Xi is an unbiased estimator of 1= ?
(Hints: if U s ( 1; ) and V s ( 2; ) and they are in-
dependent, then U + V s ( 1 + 2; ). For every > 1,
( )= ( 1) = 1
1 2
2. Let Y1;:::;Yn be a random sample from the distribution with pdf
1 2’(y #1) +1
2’(y #2), where ’(y) is the density of the standardnormal distribution. Find a method of moments estimator for #
1 and#2.34
3. Letc1;:::;cn be known times (the censoring variables). SupposeY 1 ;:::;Y n
are i.i.d. , each has an exponential distribution, i.e., it has the density
#e #y. However, we don’t observe Y i but Yi. The latter is equal to Y i
if Y i ci. If Y i >ci, then Yi ci has an exponential distribution with
parameter 1. Let NU be the number of observation with Yis smaller
than ci, and Nc with Yi >ci.
(a) Show thatYi has the density#e #yi1I(0 yi ci)+e #ci yi+ci1I(yi >
ci).
(b) Show that the MLE equals to NUPn
i=1 minfYi;cig