Information Matrix

I wrote earlier that deviance residuals were better than Pearson residuals when examining a model fit for Poisson counts. It is worth expanding on why this is, since it also neatly illustrates why there are limits to models based on grouped counts.

When fitting a model for Poisson counts, an important step is to check the goodness of fit using the following statistic:

\[\tilde{\chi}^2 = \sum_{i=1}^n r_i^2\]

The survivor function from age \(x\) to age \(x+t\), denoted \({}_tp_x\) by actuaries, is a useful tool in mortality work. As mentioned in one of our earliest blogs, a basic feature is that the expected time lived is the area under the survival curve, i.e. the integral of \({}_tp_x\). This is easy to express in visual terms, but it often requires numerical integration if there is no closed-form expression for the integral of the survival curve.

Public health initiatives, such as those being considered in the UK around sugar, carry risks as well as potential benefits for any government. The first consequence of action is the near-certain accusation of presiding over a nanny state.

Heligman and Pollard published a famous paper in 1980 with the title "The age pattern of mortality". In their paper they proposed an additive, three-component model of mortality:

\[q_x/p_x = f_I(x) + f_S(x) + f_A(x)\]