Information Matrix

Assume we have a random variable, \(X\), with expected value \(\eta\) and variance \(\sigma^2\). Often we find ourselves wanting to know the expected value and variance of a function of that random variable, \(f(X)\). Fortunately there are some workable approximations involving only \(\eta\), \(\sigma^2\) and the derivatives of \(f\). In both cases we make use of a Taylor-series expansion of \(f(X)\) around \(\eta\):

\[f(X)=\sum_{n=0}^\infty \frac{f^{(n)}(\eta)}{n!}(X-\eta)^n\]

Our new book, Modelling Mortality with Actuarial Applications, describes several non-parametric estimators of two quantities:

[Regular visitors to our blog will have guessed from the title that this posting is about obesity. If you landed here looking for WWII material, you want the other Battle of the Bulge.]

"The true nature of the Poisson distribution will become apparent only in connection with the theory of stochastic processes\(\ldots\)"

Feller (1950)

Stephen's earlier blog explained the origin of the very useful result relating the life-table survival probability \({}_tp_x\) and the hazard rate \(\mu_{x+t}\), namely:

\[ {}_tp_x = \exp \left( - \int_0^t \mu_{x+s} \, ds \right). \qquad (1) \]

To complete the picture, we add the assumption that the future lifetime of a person now aged \(x\) is a random variable, denoted by \(T_x\), and the connection with expression (1) which is:

In recent years we have published a number of papers on stochastic mortality models. A particular focus has been on the application of such models to longevity trend risk in a one-year, value-at-risk (VaR) framework for Solvency II. However, while a small group of models has been common to each paper, there have been changes in the calculation basis, most obviously where updated data have been used.