SOA Exam ASTAM Sample Question 19
Study guide: Exam ASTAM
You are fitting a lognormal distribution to a sample of size (as printed) with complete individual data.
Part a
Show that \hat{\mu} = \frac{1}{n}\sum_{j=1}^{n} ln(x_j)
Part b
Show that \hat{\sigma}^2 = \frac{1}{n}\sum_{j=1}^{n}\left(ln(x_j) - \mu\right)^2
Part c
Show that the estimate of the Covariance matrix of the lognormal parameters is
\widehat{Cov}(\hat{\mu}, \hat{\sigma}) = \begin{vmatrix} \frac{\hat{\sigma}^2}{n} & 0 \\ 0 & \frac{\hat{\sigma}^2}{2n} \end{vmatrix}Part d
Use the delta method to estimate the variance of mean of the population based on the lognormal model. State your answer in terms of the parameter estimates \hat{\mu} and \hat{\sigma}.
Answer this question and see the worked solution