Information Matrix

An important quantity in mathematical statistics is the moment of a distribution, i.e. the expected value of a given power of the observations. Moments can be either raw, centred about a particular value or standardised in some way. The simplest example is the mean of a distribution: this is the raw first moment, i.e. the expected value of each observation raised to the power 1:

Predicting the exact impact of weight upon mortality has proven to be a tricky business. That obesity is on the rise is universally acknowledged, but in recent years we have seen research studies reach differing conclusions, depending on the populations examined and the measures used.

In a previous posting I looked at how using a well founded statistical model can improve the accuracy of estimated mortality rates. We saw how the relative uncertainty for the estimate of \(\log \mu_{75.5}\) could be reduced from 20.5% to 3.9% by using a simple two-parameter Gompertz model:

\(\log \mu_x = \alpha + \beta x\qquad (1)\)

The CMI is the part of the UK actuarial profession which collates mortality data from UK life offices and pension consultants. Amongst its many outputs is an Excel spreadsheet used for setting deterministic mortality forecasts. This spreadsheet is in widespread use throughout the UK at the time of writing, not least for the published reserves for most insurers and pension schemes.

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:

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