Information Matrix

Examining residuals is a key aspect of testing a model's fit. In two previous blogs I first introduced two competing definitions of a residual for a grouped count, while later I showed how deviance residuals were superior to the older-style Pearson residuals. If a model is correct, then the deviance residuals by age should look like random N(0,1) variables.

A mathematical model that obtains extensive and useful results from the fewest and weakest assumptions possible is a compelling example of the art. A survival model is a case in point. The only material assumption we make is the existence of a hazard rate, \(\mu_{x+t}\), a function of age \(x+t\) such that the probability of death in a short time \(dt\) after age \(x+t\), denoted by \({}_{dt}q_{x+t}\), is:

\[{}_{dt}q_{x+t} = \mu_{x+t}dt + o(dt)\qquad (1)\]

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.]