Actuarial Notes

SOA Exam ASTAM Spring 2023, Question 2

Study guide: Exam ASTAM

You are given the following claim frequency data for a line of business with 100 one-year policies.

Number of Claims ( $k$ )	0	1	2	3	4	5+
Policies ( $n_k$ )	46	34	13	4	3	0

You are particularly interested in the probability that a policy will have no claims in a year, denoted $p_0$ .

You first fit a Poisson ( $\lambda$ ) distribution to this data using Maximum Likelihood Estimation (MLE).

Part a (3 points)

You are given that the MLE of $\lambda$ is $\hat{\lambda} = 0.84$ .

(i) (1 point) Show that the MLE of $p_0$ is $\hat{p}_0 = 0.43$ to the nearest 0.01. You should calculate the value to the nearest 0.001.
(ii) (2 points) Use the delta method to construct an approximate 95% linear confidence interval for $p_0$ under this model.

Part b (3 points)

You next fit a zero-modified Poisson distribution to this data.

(i) (1 point) Write down the likelihood function for the zero-modified Poisson distribution in terms of the model parameters ( $\lambda$ and $p_0^M$ ) and the data ( $n_k$ ).
(ii) (2 points) Using the likelihood function, show that the MLE of $p_0^M$ is $\hat{p}_0^M = 0.5$ to the nearest 0.1. You should calculate the value to the nearest 0.01.

Part c (2 points)

(i) Show that $\mathrm{Var}\left[\hat{p}_0^M\right] = \dfrac{p_0^M\left(1 - p_0^M\right)}{100}$ .
(ii) Construct an approximate 95% linear confidence interval for $p_0^M$ under this model.

Part d (2 points)

You have calculated the following maximum log-likelihood values for the two models:

Model	Poisson	Zero-Modified Poisson
Maximum Log-likelihood	-124.36	-123.91

(i) Calculate the Schwarz Bayes Criterion (SBC) for both models.
(ii) Your colleague claims that, because the SBC favors the Poisson model, the Poisson model will be superior for predicting the value of $p_0$ . State with reasons whether or not your colleague is correct.

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