SOA Exam ASTAM Sample Question 25

Study guide: Exam ASTAM

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with Pareto distribution having unknown parameter $\alpha$ but known parameter $\theta = 100$ .

Part a

Show that the maximum likelihood estimate (MLE) of $\alpha$ can be expressed as

 $\hat{\alpha} = \frac{n}{\sum_{i=1}^{n}\log(x_i + 100) - n\log(100)}.$

Part b

Determine the asymptotic variance of the MLE of $\alpha$ .

Part c

Give a general expression for a 95% confidence interval for $\alpha$ .

Part d

Let the mean of $X$ be $\mu$ .

(i) Express the MLE of $\mu$ in terms of the MLE of $\alpha$ .

(ii) Obtain its asymptotic variance.

(iii) Give a general expression for a 95% confidence interval for $\mu$ .

Answer this question and see the worked solution

Concepts tested

Maximum Likelihood Estimation
Pareto Distribution
Fisher Information
Approximate Confidence Interval
Delta Method