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Posts feedWhen is your Poisson model not a Poisson model?
The short answer for mortality work is that your Poisson model is never truly Poisson. The longer answer is that the true distribution has a similar likelihood, so you will get the same answer from treating it like Poisson. Your model is pseudo-Poisson, but not actually Poisson.
See You Later, Indicator
A recurring feature in my previous blogs, such as this one on information, is the indicator process:
\[Y^*(x)=\begin{cases}1\quad\mbox{ if a person is alive at age \(x^-\)}\\0\quad\mbox{ otherwise}\end{cases}\]
where \(x^-\) means immediately before age \(x\) (never mind the asterisk for now). When something keeps cropping up in any branch of mathematics or statistics, there are usually good reasons, and this is no exception. Here are some:
Less is More: when weakness is a strength
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:
\[{}_{dt}q_{x+t} = \mu_{x+t}dt + o(dt)\qquad (1)\]
Stopping the clock on the Poisson process
"The true nature of the Poisson distribution will become apparent only in connection with the theory of stochastic processes\(\ldots\)"
Feller (1950)
Everything points to Poisson
One recurring theme in our forthcoming book, Modelling Mortality with Actuarial Applications, is the all-pervading role of likelihoods that suggest the lurking presence of a Poisson distribution. A popular assumption in modelling hazard rates is that the number of deaths observed at any given age is a Poisson random variable, so perhaps that might explain it?
Minding our P's, Q's and R's
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:
\[\tilde{\chi}^2 = \sum_{i=1}^n r_i^2\]