SOA Exam ASTAM Sample Question 10

For a portfolio of short-term insurance contracts, claim amounts are modeled with a density function

$f(x; \lambda) = \lambda^2 x e^{-\lambda x} \qquad x > 0,\ \lambda > 0$

A random sample of $n$ claim amounts, $x_1, x_2, \cdots, x_n$ , is used to estimate the parameter $\lambda$ .

Part a

Derive and simplify as far as possible a formula for the maximum likelihood estimate of $\lambda$ .

Use the moment generating function to show that $Y = 2n\lambda\bar{X}$ has a chi-square distribution and state the degrees of freedom.

Determine the Fisher information function, $I(\lambda)$ and use it to derive a formula for the asymptotic variance of $\hat{\lambda}$ .

A random sample of 10 claim amounts resulted in a sample mean of $\bar{x} = 358$ .

(i) Use the results in (b) to calculate an exact 95% confidence interval of $\lambda$ .
(ii) Use the results in (c) to calculate an approximate (asymptotic) 95% confidence interval of $\lambda$ .

Comment briefly on the comparison of these confidence intervals.