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This is the Past Exam of Math Tripos which includes Topics in Representation Theory, Topics in Group Theory, Time Series and Monte Carlo Inference, Three-Dimensional Manifolds etc. Key important points are: Statistical Theory, D-Dimensional Parameter, Consequences of Parameter, Profile Likelihood, Modified Profile Likelihood, Log-Likelihood Function, Cumulant Generating Function, Saddlepoint Approximation, M-Estimator of Parameter, M-Estimator At Distribution
Typology: Exams
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Thursday 7 June 2001 1.30 to 4.
Attempt any FOUR questions. The questions carry equal weight.
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.
Paper 32
