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A past exam paper for the m.phil. In statistical science program, focusing on statistical theory. It includes instructions for the exam, as well as questions covering topics such as profile likelihood, conditional likelihood, transformation models, minimal sufficient statistics, m-estimators, and bayesian decision theory. Candidates are required to answer questions related to these topics and demonstrate a deep understanding of statistical concepts.
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Thursday 2 June 2005 1.30 to 4.
Attempt FOUR questions, not more than TWO of which should be from Section B.
There are TEN questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
Section A
1 Explain briefly the concepts of profile likelihood and conditional likelihood, for inference about a parameter of interest ψ, in the presence of a nuisance parameter λ.
Suppose Y 1 ,... , Yn are independent, identically distributed from the exponential family density f (y; ψ, λ) = exp{ψτ 1 (y) + λτ 2 (y) − d(ψ, λ) − Q(y)},
where ψ, λ are both scalar.
Obtain a saddlepoint approximation to the density of S = n−^1
∑n i=1 τ^2 (Yi). Show that use of the saddlepoint approximation leads to an approximate conditional log-likelihood function for ψ of the form
lp(ψ) + B(ψ),
where lp(ψ) is the profile log-likelihood, and B(ψ) is an adjustment which you should specify carefully.
2 Explain in detail what is meant by a transformation model.
What is meant by (i) a maximal invariant, (ii) an equivariant estimator, in the context of a transformation model?
Describe in detail how an equivariant estimator can be used to construct a maximal invariant. Illustrate the construction for the case of a location-scale model.
5 Let Y 1 ,... , Yn be independent, identically distributed from a distribution F , with density f symmetric about an unknown point θ. Suppose we wish to test H 0 : θ = θ 0 against H 1 : θ < θ 0.
Explain how to test H 0 against H 1 using (i) the sign test, and (ii) the Wilcoxon signed rank test.
Show that the null mean and variance of the Wilcoxon signed rank statistic are 1 4 n(n^ + 1) and^
1 24 n(n^ + 1)(2n^ + 1) respectively. What is meant by a one-sample U -statistic? State, without proof, a result concerning the asymptotic distribution of a one- sample U -statistic, and use it to deduce asymptotic normality of the Wilcoxon signed rank statistic.
6 Write brief notes on four of the following:
(i) Edgeworth expansion;
(ii) parameter orthogonality; (iii) Laplace approximation;
(iv) Bartlett correction;
(v) the invariance principle; (vi) finite-sample versions of robustness measures;
(vii) tests based on the empirical distribution function; (viii) large-sample likelihood theory.
Section B
7 Assume that the n-dimensional observation vector Y may be written
Y = Xβ + ,
where X is a given n × p matrix of rank p, β is an unknown vector, and
∼ Nn(0, σ^2 I).
Let Q(β) = (Y − Xβ)T^ (Y − Xβ). Find βˆ, the least-squares estimator of β, and show that
Q( βˆ) = Y T^ (I − H)Y
where H is a matrix that you should define.
If now Xβ is written as Xβ = X 1 β 1 + X 2 β 2 , where X = (X 1 : X 2 ), βT^ = (βT 1 : β 2 T ), and β 2 is of dimension p 2 , state without proof the form of the F -test for testing H 0 : β 2 = 0.
What is meant by saying that β 1 is orthogonal to β 2? What is the practical relevance of orthogonality?
8 Suppose that Y 1 ,... , Yn are independent binomial observations, with
Yi ∼ B(ti, πi) and log(πi/(1 − πi)) = βT^ xi, for 1 6 i 6 n,
where t 1 ,... , tn and x 1 ,... , xn are given. Discuss carefully the estimation of β.
Your solution should include
(i) the method of checking the fit of the above logistic model,
and
(ii) the method for finding an approximate 95% confidence interval for β 2 , the second component of the vector β.
