MATH 374 STATISTICAL METHODS IN INSURANCE & FINANCE FINAL EXAM 2022
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MATH 374
FINAL EXAM 2022
STATISTICAL METHODS IN INSURANCE & FINANCE
1. Consider the following loss function
L(θ, d) = - log ╱ 、 - 1.
Let X1 , . . . , Xn be exchangeable so that the Xi are conditionally independent given parameter θ . Suppose that Xi | θ ~ Poisson(θ) and θ ~ Gamma(α, 1) with α > 1.
Find the Bayes rule of an immediate decision when π(θ) is the prior distribution. [10 marks]
2. The number of claims in a year has a Poisson distribution with mean λ . The parameter λ has the uniform distribution over the interval (1, 10).
(i). If an insured had one claim during the first year, estimate the expected number of claims for the second year using B¨uhlmann credibility. [5 marks]
(ii). If an insured had one claim during the first year, estimate the expected
number of claims for the second year using Bayesian credibility. [5 marks]
3. Let {Xi | θ}i≥1 be normally distributed with mean µ (known) and variance ϑ .
(i). Show that f (xi | ϑ) belongs to the exponential family and for X = (X1 , . . . , Xn ) state the sufficient statistics for learning about ϑ . [10 marks]
(ii). Write down the likelihood for X = (X1 , . . . , Xn ). By viewing the likelihood
as a function of ϑ , which generic family of distributions (over ϑ) is the likelihood a kernel of? [10 marks]
(iii). Let ϑ be inverse Gamma with parameters a and b, i.e. ϑ ~ InvG(a, b).
By first finding the corresponding posterior distribution for ϑ given x = (x1 , . . . , xn ), show that this family of distributions is conjugate with respect
to the likelihood f (x | ϑ). [10 marks]
2023-05-26