SOA Exam ASTAM Fall 2025, Question 2

Study guide: Exam ASTAM

You are analyzing the distribution of the number of claims during a one-year period for a portfolio of 2000 property insurance policies. The number of claims for each policy are assumed to be independent and identically distributed. You are given the following observed data:

You first fit the data using a negative binomial distribution with r = 2, and unknown \beta.

You are given that the maximum likelihood estimate of \beta is \hat{\beta} = 0.21825.

Part a (2.5 points)

Show that the maximum value of the log-likelihood function under this model is -1747 to the nearest 1. You should calculate the value to the nearest 0.01.

Part b (2.5 points)

You conduct a chi-square goodness of fit test with 5 cells at the 5% significance level.

(i) Show that the test statistic for this test is 14 to the nearest 1. You should calculate the value to the nearest 0.1.

(ii) Determine the critical value for this test.

(iii) State the conclusion of this test.

Part c (3 points)

You next consider an alternative negative binomial model with both r and \beta estimated by maximum likelihood. You are given the following information.

• The maximum likelihood estimates are \hat{r} = 3.814, \hat{\beta} = 0.114.
• The value of the log-likelihood function with these parameters is -1744.43.
• The test statistic for a chi-square goodness of fit test with 5 cells is 12.982.

State with reasons which of the two models is preferred based on each of the following criteria:

(i) The Akaike Information Criterion (AIC).

(ii) The Schwartz Bayes Criterion (SBC/BIC).

(iii) The likelihood ratio test at the 5% significance level.

(iv) The chi-square goodness of fit test.

Part d (1 point)

Your colleague suggests using a zero-modified negative binomial instead of either of the models used above. Explain why this suggestion is unlikely to significantly improve the chi-square goodness of fit of the negative binomial distribution for these data.

Answer this question and see the worked solution

Concepts tested

• Chi-Square Goodness-of-Fit Test
• Likelihood Ratio Test
• AIC
• BIC
• Negative Binomial Distribution
• Maximum Likelihood Estimation

More questions from this set

• SOA Exam ASTAM Fall 2025, Question 3