2016-17 Poge2of7
Section A (60 marks)
1. Let X1,X2, . . . ,Xn be independent and identically distributed random variables.
(a) Suppose that the distribution is given by a probability density function/probability
function f (:L’,0), where 6 E (-3 is a real parameter. Define the maximum likelihood
estimator of 0 based on the sample X1,X2,. . . ,Xn.
(b) Suppose that the common probability density function is
404
f(x,6) = E for x Z 0,
where 0 > 0. Find the maximum likelihood estimator of the parameter 0.
(8 marks)
2. Let X1, X2, . . . ,Xn be independent and identically distributed random variables with a com-
mon distribution depending on an unknown parameter 6.
(a) Explain the method of moments to estimate the parameter (9.
(b) Suppose that the common probability density function is the same as in 1(b).
(i) Find a moment estimator of the parameter 6.
(ii) Is the estimator in (i) unbiased? Justify your answer.
(10 marks)
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MT3320 Page 3 of 7
3. Let X1,X2,. . . ,Xn be independent and identically distributed random variables with a com-
mon probability density function/probability function f (r, 6).
(a) State, without proof, the Cramér—Rao theorem for an unbiased estimator 9(X) of g(6),
where X = (X1,X2,…,X,,) and 9(6) is a differentiable function of the parameter 6.
(b) Suppose that the common distribution is Gamma I‘(4, 6), that is,
3
:1: an
f(.r,6) — 664 exp for x >0,
where 6 > 0. Find the Cramér-Rao lower bound for the variance of an unbiased esti-
mater of 6.
(Assume that the regularity conditions are satisfied.)
(12 marks)
4. Let X1,X2, . . . ,Xn be independent and identically distributed random variables with a com-
mon probability density function/ probability function f(:z:, 6).
(a) State, without proof, the theorem about the attainment of the Cramér-Rao lower
bound.
(b) Suppose that the common distribution is Bernoulli B(1, 6). Using the theorem about the
attainment of the Cramér—Rao lower bound, identify the minimum variance unbiased
estimator for 6.
(Assume that the regularity conditions are satisfied.)
(10 marks)
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2016—1 7 Page A of 7
5. Let X1, X2, . . . , Xn be random variables with a joint probability density function/probability
function f(x, 6) = f(m1,z2,…,xn,6),whorc 6 E G.
(a) Explain what is meant by saying that C is a test of size a.
(b) State. without proof, the Neyman-Pearson fundamental lemma.
(5 marks)
6. Let X1,X2,…,X,. be independent and identically distributed random variables with the
common probability density function
f(z,6) = 6exp(—6:r) for a: > 0,
where 6 > O.
(a) Derive the Neyman-Pearson critical region for the test of the null hypothesis
H0 : 6 = 60 against the alternative H, z 6 = 61, with 60 < 61.
(You may assume that 26 22:, X, has the X2(2n) distribution.)
(b) Specify the region when 60 = 2,61 = 3, n = 10, and or z 0.05, and find the power of
the test.
(15 marks)
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MT3320 Page 5 of 7
Section B (40 marks)
7. (a) Let X1,X2,...,X,, be independent and identically distributor] random variables with
a common probability density flmotion/probahility function f(:r., 6) depending on an
unknown parameter 6. Using~ the factorisation theorem (criterion), prove that any one-
to-one function of a sufficient statistic is a sufficient statistic.
(b) Suppose that the common distribution is Binomial B(N, 6), that is,
f(m,6)=
mean E(X) = A
variance Var(X) : A
mgr W) = mow — 1)}
Geometric distribution C(p)
pf f(z,p)=(1-p)“‘p;z=1,2,-~-;0
0
mean E(X): p
variance Va.r(X) = 02
mi M (t) = eXDW + 029/2}
Uniform distribution U (a, b)
pdf f(z,a,b)=@;a$ng;a O
mam E(X) = 1/9
variance Va.r(X) = 1/02
myf MU) = (1 ” M9)—1
Gamma distribution I‘(a,B)
Mi f($,a,l3)=n;lm;z°“1exp{—§};z>0; 01,3}0
mean E(X) = of?
variance Var(X) = (1,62
MU) = WV”
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END T. Sharla
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