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A past exam paper from an m.phil. In statistical science program, focusing on statistical theory. It includes questions on maximum likelihood estimation, hermite polynomials, conditional likelihood, p-formula, functional statistics, and hypothesis testing. Students are required to find maximum likelihood estimators, use edgeworth expansions, explain concepts, and derive expressions.
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Thursday 5 June 2003 9 to 12
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.
Section A 1 (i) Let Y 1 ,... , Yn be independent, identically distributed exponential random variables with common density f (y ; λ) = λe−λy^ , y > 0, and suppose that inference is required for θ = E(Y 1 ). Find the maximum likelihood estimator of θ, and explain carefully why, with Y¯ = n−^1 ∑ni=1 Yi and Φ the distribution function of N (0, 1), ( (^) ¯ Y 1 + n−^1 /^2 Φ−^1 ( 1 − α 2 )^ ,^
1 − n−^1 /^2 Φ−^1 ( 1 − α 2 )
is a confidence interval for θ of asymptotic coverage 1 − α. (ii) Define the rth^ degree Hermite polynomial Hr (x). Let X 1 ,... , Xn be independent, identically distributed random variables, with common mean μ and common variance σ^2 , and let
T =
( (^) ∑n
i=
Xi − nμ
/√nσ.
An Edgeworth expansion of the distribution function of T is
P (T 6 t) = Φ(t) − φ(t)
{ (^) ρ 3 6 √n H^2 (t) +^
ρ 4 24 n H^3 (t) +^
ρ^23 72 n H^5 (t)
in terms of standardised cumulants ρr. Use an appropriate Edgeworth expansion to show that the confidence interval (∗) in (i) above has coverage error of order O(n−^1 ).
2 Explain briefly the concept of a conditional likelihood. 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)}. Find the cumulant generating function of τ 2 (Yi), and a saddlepoint approximation to the density of S = n−^1 ∑ni=1 τ 2 (Yi). Show that the saddlepoint approximation leads to an approximation to a condi- tional log-likelihood function for ψ of the form
l(ψ, λˆψ ) + 12 log |dλλ(ψ, ˆλψ )|,
in terms of quantities ˆλψ , dλλ which you should define carefully.
Paper 41
Section B 7 Suppose (rij ) are independent observations, with
rij ∼ Bi(nij , pij ) , 1 6 i, j 6 2 ,
where n 11 , n 12 , n 21 , n 22 are given totals. Consider the model
ω : logit pij = μ + αi + βj , 1 6 i, j 6 2
where α 1 = β 1 = 0. (i) Write down the log-likelihood under ω, and discuss carefully how ˆα 2 , βˆ 2 and their corresponding standard errors may be derived. [Do not attempt to find analytical expressions for ˆα 2 , βˆ 2 and their se’s.] (ii) With (rij /nij ) as 498/796 878/ 54/142 197/
and fitting ω by glm( ), the deviance was found to be .00451, with
αˆ 2 = − 1 .013(se = .0872) βˆ 2 = − 0 .3544(se = .0804).
What do you conclude from these figures?
8 Let Y 1 ,... , Yn be independent variables, such that Y = Xβ + , where X is a given n × p matrix, of rank p, β is an unknown vector of dimension p, and 1 ,... , n are independent normal variables, each with mean 0 and unknown variance σ^2. (i) Derive an expression for βˆ, the least squares estimator of β, and derive the distribution of βˆ. (ii) How would you estimate σ^2? (iii) In fitting the model Yi = μ + αxi + βzi + γti + i , 1 6 i 6 n, where (xi), (zi), (ti) are given vectors, and 1 ,... , n has the distribution given above, explain carefully how you would test H 0 : β = γ = 0.
(You may quote any standard theorems needed.)
Paper 41
9 Write an account, with appropriate examples, of the decision theory approach to inference. Your account should include discussion of all of the following: (i) the main elements of a decision theory problem; (ii) the Bayes and minimax principles; (iii) admissibility; (iv) finite decision problems; (v) decision theory approaches to point estimation and hypothesis testing.
10 Suppose that Y 1 and Y 2 are independent Poisson random variables with means ψμ and μ respectively. We are interested in testing the null hypothesis H 0 : ψ 6 ψ 0 against the alternative hypothesis H 1 : ψ > ψ 0 , where ψ 0 is given and μ is unknown. Explain in detail why the appropriate test is a conditional test, based on the conditional distribution of Y 1 given Y 1 + Y 2 , and find its form. Let S = Y 1 + Y 2. Show that the significance probability for observed (Y 1 , Y 2 ) is approximately 1 − Φ
[ (^) Y 1 − Sψ 0 /(1 + ψ 0 ) {Sψ 0 /(1 + ψ 0 )^2 }^1 /^2
Paper 41
