ST903: Statistical Methods Assignment 2
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ST903: Statistical Methods
Assignment 2
1: Let X1 , . . . ,Xn be i.i.d. with the pdf
f(x;θ) = θxθ − 1 ,
for 0 ≤ x ≤ 1 and θ > 0.
(i) Find the maximum likelihood estimator (MLE) θˆ of θ . (Help: do not forget that this includes verifying that θˆ is indeed MLE). [3]
(ii) If we define Yi = −log Xi , then derive the distribution of Yi and of 对
Yi . In each case
give the name of the distribution and clearly state the relevant parameters. [3]
(iii) Derive the first two moments, i.e. the expectation and the variance, of θˆ. (iv) Is your MLE θˆ a consistent estimator of θ? Explain your answer.
[3]
[2]
[TOTAL: 11]
2: In order to model counts of accidents on a stretch of road, we consider the following hierarchical
Bayesian model:
We model the counts Y1 , . . . ,Yn as independently generated from a Poisson distribution with rate parameter λ , i. e. for i = 1, . . . ,n:
exp(−λ)λy
P(Yi = y|λ) = y! , y = 0, 1, 2..
We further assume that λ has a Gamma(α,β) prior distribution which corresponds to the probability density function
f(λ|β) = βα Γ(α)− 1 λα − 1 exp(−βλ) , λ > 0,
where α is a known positive number while the scale parameter in this prior distribution, β, is itself unknown and has a Gamma(a,b) prior with known a,b > 0.
(i) Write down the likelihood for λ given the observed sample y1 , . . . ,yn . [2]
(ii) Derive the joint posterior distribution of the free parameters λ,β given y1 , . . . ,yn (up
to a proportionality constant) and explain why a numerical method is required here for conducting inference from this posterior. [3]
(iii) Derive the relevant full conditionals and explain how you would analyse the posterior
distribution obtained in (ii) through Gibbs sampling. [4]
[TOTAL: 9]
2022-12-09