SOA Exam ASTAM Spring 2024, Question 4

The loss severity for an insurance policy has the following density function:

$f(x) = \frac{3x^2 e^{-(x/\theta)^3}}{\theta^3} \qquad \text{for } x > 0$

For this distribution, the first 3 raw moments are:

$\mathrm{E}[X] = \theta\,\Gamma\!\left(1 + \frac{1}{3}\right) \qquad \mathrm{E}\big[X^2\big] = \theta^2\,\Gamma\!\left(1 + \frac{2}{3}\right) \qquad \mathrm{E}\big[X^3\big] = \theta^3$

(Note: You can use the GAMMA function in Excel.)

A random sample of 100 losses, $x_1, x_2, \cdots, x_{100}$ , has been collected. You are given:

$\sum_{i=1}^{100} x_i = 954 \qquad \sum_{i=1}^{100} x_i^2 = 9{,}888 \qquad \sum_{i=1}^{100} x_i^3 = 109{,}812$

Part a (2 points)

Determine the log-likelihood function for this sample.

Part b (2 points)

Show that the maximum likelihood estimate of $\theta$ , denoted $\hat\theta$ , is 10.3 to the nearest 0.1. You should calculate the value to the nearest 0.0001.

Part c (2 points)

Use Fisher's information to estimate the standard deviation of $\hat\theta$ .

Part d (2 points)

(i) Calculate the maximum likelihood estimate of $\mathrm{Var}[X]$ .

(ii) Using the delta method, estimate a 95% confidence interval for $\mathrm{Var}[X]$ .

Part e (2 points)

Maximum likelihood estimates are known to have the following characteristics:

A. Invariance property

B. Asymptotically normally distributed

(i) Briefly describe each of these characteristics.

(ii) Explain where each of these characteristics has been applied in this question.

Answer this question and see the worked solution