SOA Exam ASTAM Sample Question 11

Study guide: Exam ASTAM

For a portfolio of 2000 policyholders of automobile insurance, you are given the observed number of claims per policyholder for a given period:

Part a

Assume first that N, can be modeled as a Poisson random variable.

• (i) Calculate the maximum likelihood estimate of the Poisson parameter.
• (ii) Calculate the expected number of policyholders for each possible number of claims using the fitted Poisson distribution.
• (iii) Calculate the chi-square test statistic for testing the goodness-of-fit of the Poisson model to the observed data.

Part b

Now assume that the distribution of N is given by

\Pr\left[N = n\right] = \pi(1-\pi)^n \qquad n = 0, 1, 2, \cdots \qquad 0 < \pi < 1

• (i) Calculate the maximum likelihood estimate of \pi.
• (ii) Calculate the expected number of policyholders for each possible number of claims using the fitted distribution.
• (iii) Calculate the chi-square test statistic for testing the goodness-of-fit of the given model to the observed data.

Part c

Compare the suitability of the two models to the data.

Answer this question and see the worked solution

Concepts tested

• Maximum Likelihood Estimation
• Chi-Square Goodness-of-Fit Test
• Poisson Distribution
• Geometric Distribution
• Hypothesis Testing

More questions from this set

• SOA Exam ASTAM Sample Question 10
• SOA Exam ASTAM Sample Question 12