程序代写代做代考 2016-17 Poge2of7

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) No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. NEXT PAGE 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)=0

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