2014—15 Poge2ofo
1. Let X1, X2, . . . , X,, be independent random variables with a common distribution depending
on an unknown parameter 9 E (9.
(a) (i) Explain what is meant by saying that is an unbiased estimator of a real-
valued function g(9) of the parameter 9.
(ii) Is the following true or false? You do NOT need to justify your answer.
(1) If 9 is a maximum likelihood estimator of the parameter 9, then 9 is unbi-
ased.
(2) If 9 is a maximum likelihood estimator of the parameter 9, then 9 is efficient.
(3) If 9 is a maximum likelihood estimator of the parameter 9, then 9 is asymp-
totically efficient.
(4) If 9 is a moment estimator of the parameter 9, then 9 is unbiased.
(5) If 9 is a moment estimator of the parameter 9, then 9 is efficient.
(6) If 9 is a moment estimator of the parameter 9, then 9 is asymptotically
efficient.
(7) Suppose that 9 is the maximum likelihood estimator of 9 and 9(9) is a
real-valued function of 9. Then 9(9) is the maximum likelihood estimator of
9(0)-
(b) Suppose that the common distribution is the Gamma F(2, 9), that is, the common
probability density function is
f(a:, 9) = $3315— mm for :1:>0,
where 9 > O.
(i) Find the maximum likelihood estimator 9 of the parameter 9.
(ii) Is the estimator in (i) unbiased? Justify your answer.
(c) Suppose that the common probability density function is
f(:r, 9) = 9(9 + 1)x9_1(1— 2:) for O < a: < 1, where 9 > 0. Find a moment estimator of the parameter 9.
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MT3320 Page 3 of (S 2 I l-JIOLLOWAY
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2. Let X1, ‘2,…,X,, be independent random variables with a common probability density
function/probability function f(a:, 9).
(a) (i) Give the dehnitions of a sufficient statistic and a minimal sufficient statistic.
(ii) State, without proof, the Rae-Blackwell theorem.
(iii) Given a complete sufficient statistic T for 9, explain how to obtain a minimum
variance unbiased estimator g) for a real-valued function 9(9).
(You need not define the notion of completeness.)
(b) Suppose that
f(x, 9) = 9225:?” for :1: > 0,
where 9 > O.
(i) By writing the probability density function in exponential family form, show
that T = ZLI X,- is a complete sufficient statistic for 9.
(ii) Prove that 9 = 1/X1 is an unbiased estimator of 9.
(iii) Assuming that the conditional probability density function of X1 given T = t is
fx,lT(x|t) = (2n — 1)(2n — 2)t,ff_1 (t — z)2″‘3 for 0 S x g t,
show that
(211. — 1)
21″;1 Xi
is the minimum variance unbiased estimator of 9.
é:
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2014—l5 Page A of 6
3. Let X1,X2,…,X,, be independent random variables with a common probability density
function/probability function f(x, 9).
(a) (i) Define the one-step Fisher information and the total Fisher information.
(ii) Assume that the regularity conditions of the Cramér-Rao Theorem hold. Give,
without proof, an alternative formula for the one-step Fisher information.
(iii) State, without proof, the theorem about the attainment of the Cramér—Rao lower
bound.
(You. need not list the regularity conditions.)
(b) Suppose that the common distribution is the Gamma I‘(4, 9), that is
3
m z
9 = _ —— .NI.) 604exp( 0) for x>0and9>0
(i) Find the Cramér—Rao lower bound for the variance of an unbiased estimator of
9.
(Assume that the regularity conditions are satisfied.)
(ii) Using the theorem about the attainment of the Cramér—Rao lower bound, identify
the minimum variance unbiased estimator for 9.
(Assume that the regularity conditions are satisfied.)
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MT3320 Page 5 of 6 -. “QEkwAV
4. (a) Let X1, 1 ’2, . . . , X,, be random variables with ajoint probability density function/probability
function f(x, 9) = f(x1,x2,…,zn,9),whcrc 9 E E).
(i) Define the power function of a test C.
(ii) Suppose that C is a test for a null hypothesis Ho : 9 E 80 against the alternative
hypothesis H1 : 9 E 91, where 61 = G\@0. Explain what is meant by saying
that the test C is unbiased.
(iii) State, without proof, the Neyman—Pearson fundamental lemma for the test of a
simple hypothesis H0 : 9 = 90 against a simple alternative H1 : 9 = 91.
(b) Let X1,X2,.. be a random sample from the normal distribution N(0,02).
(i) Obtain the Neyman—Pearson critical region of size oz for the test of the null
hypothesis H0 : a = 00 against the alternative H1 : a = 01 where 01 > 00.
(You may assume that 0% EZ=1X3 has the X2(n) distribution.)
(ii) Specify the critical region when n = 21, 00 = 1.2, 01 = 1.8 and a = 0.01, and
find the power.
5. (a) Explain the application of the likelihood ratio test to composite hypotheses.
(b) Let X1,X2,. . . , X” be a random sample from the normal distribution N (u, 02), where
both [i and 0’ are unknown. Suppose that C is the critical region for the likelihood ratio
test for testing the null hypothesis Ho : u = #0 against the alternative H1 2 u 75 #0-
(i) Show that C can be written as
where x = (zl,z2,…,z,,), 6’2 = §ZS=,(I,- — if, 63 = i Z; u0)2, 5: =
% 221:1
(ii) Show that the critical region of size a can be specified using tables of the t-
distribution. /2
7).
(You may assume that, if A S 1, S ,\ is equivalent to 520.02 2
if—x—z/n_ 1.)
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MT3320 Page 7 of 7
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Table of Standard Distributions
Binomial distribution B(n,p)
Pf Hump):(n)P“’(1-p)””;x=0,1….,n;0
mean E(X) : /\
variance Van-(X) = A
mgf 1i1(t) = e.xp{)\(et — 1)}
Geometric distribution G(p)
pf f(z,p)=(1—P)I_1P§$=1i2:—§0
mean E(X) = a
variance Va.r(X) = 02
mgf M (t) = expwt + U2t2/2}
Uniform distribution U (a, b)
pdf f(x,a,b)=@;a$rcgb;a 0
mean E(X) = 1/9
variance Va.r(X) = 1/02
myf MU): (1 — U9)—1
Gamma distribution l”(a, B)
Pdf f($,0al9)=al$j$”_lexp{—% ;z>0;a,;9>0
mean E(X) = 041
variance Var(X) = cw”
mgr Ma) = (1 — a)”
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END T. Sharla