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ACST840 Quantitative Research Methods

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Assume that the corresponding annual number of loss events Nt , t = 1, ..., 4 are independent and identically distributed from P oisson(λ) and severities Xi are independent and identically distributed from lognormal distribution LN (µ, σ2). Also Nt and Xi are all independent.
a) Estimate λ, µ, σ using the maximum likelihood method.
b) Using Markov chain Monte Carlo method (random walk Metropolis-Hastings within Gibbs
algorithm studied during the course) with 104
chain iterations, under the Bayesian approach
with constant priors, estimate the posterior mean and the posterior standard deviation for
λ, µ, σ.
c) Using closed form solution, under the Bayesian approach with constant priors, estimate the
posterior mean and posterior standard deviation for λ.
d) Under the Bayesian approach, model parameter λ is modelled by random variable Λ. Suppose that expert’s best estimates for the expected number of annual losses and associated
uncertainty are E[Λ] = 1 and Vco[Λ] = 0.5. If the prior distribution for Λ is Gamma(α, β),
find α and β using expert estimates and calculate (using closed form solution and MCMC)
the posterior mean and posterior standard deviation for Λ.
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