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A past exam paper for a master's degree in statistical science, focusing on statistical theory. It includes questions on orthogonality of parameters, profile likelihood, saddlepoint approximation, m-estimators, maximal invariants, and diagnostic checks. Students are required to attempt four questions, with no more than two from section b.
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Thursday 7 June 2001 1.30 to 4.
You should attempt FOUR questions, no more than two of which should be from Section B.
1 Let a d-dimensional parameter vector θ be partitioned as θ = (ψ, λ).
Explain what is meant by orthogonality of ψ and λ. Discuss briefly the consequences of parameter orthogonality for maximum likelihood estimation.
Suppose that Y is distributed according to a density of the form
PY (y; θ) = a(λ, y)exp{λt(y; ψ)}.
Show that ψ and λ are orthogonal.
2 Write a brief account of the concept and properties of profile likelihood.
Define what is meant by modified profile likelihood.
Let Y 1 ,... , Yn be independent, identically distributed according to an inverse Gaussian distribution with density
{ψ/(2πy^3 )}^1 /^2 exp {−
ψ 2 λ^2 y
(y − λ)^2 }, y > 0
where ψ > 0 and λ > 0. The parameter of interest is ψ.
Find the form of the profile log-likelihood function and of the modified profile log- likelihood.
(^3) (i) Let Y 1 ,... , Yn be independent, identically distributed random variables with den- sity fY (y) and cumulant generating function KY (t). Describe in detail the saddlepoint approximation to the density of
Y = n−^1
∑^ n
i=
Yi.
(ii) Let Y 1 ,... , Yn be independent random variables each with a Laplace density
fY (y) = exp{−|y|}/ 2 , −∞ < y < ∞.
Show that the cumulant generating function is KY (t) = −log(1 − t^2 ), |t| < 1, and derive the form of the saddlepoint approximation to the density of Y.
7 (i) Assume that the n-dimensional observation vector Y may be written
Ω : 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). Show that Q(β) is a convex function of β, and find βˆ, the least-squares estimator of β. Show also that
Q( βˆ) = Y T^ (I − H)Y
where H is a matrix that you should define.
(ii) Let ˆ = Y − X βˆ. Find the distribution of ˆ, and discuss how this may be used to perform diagnostic checks of Ω.
(iii) Suppose that your data actually corresponded to the model
Yi ∼ N (μi, σ^2 i ), 1 ≤ i ≤ n, with σ^2 i ∝ μ^2 i.
How would your diagnostic checks detect this, and what transformation of Yi would be appropriate?
8 Suppose that Y 1 , · · · , Yn are independent Poisson random variables, with E(Yi) = μiti, 1 ≤ i ≤ n, where t 1 , · · · , tn are given times. Discuss carefully how to fit the model
H 0 : logμi = βT^ xi, 1 ≤ i ≤ n,
where x 1 , · · · , xn are given covariates, and β is a vector of unknown parameters.
9 Write a brief account of the role of conditioning in classical statistical inference.
Contrast briefly the handling of nuisance parameters in classical approaches to inference with that in the Bayesian approach.
10 Let X 1 ,... , Xn be independent, identically distributed random variables, with the exponential density f (x; θ) = θe−θx, x > 0.
Obtain the maximum likelihood estimator θˆ of θ. What is the asymptotic distribution of
n(θˆ − θ)?
Show that θˆ is biased as an estimator of θ.
What is the minimum variance unbiased estimator of θ? Justify your answer carefully, stating clearly any results you use.
