Stopping the clock on the Poisson process

(Apr 12, 2018)

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

Feller (1950)

 

In a previous blog, we showed how survival data lead inexorably toward a Poisson-like likelihood. This explains the common assumption that if we observe \(D_x\) deaths among \(n\) individuals, given \(E_x^c\) person-years exposed-to-risk, and we assume a constant hazard rate \(\mu\), then \(D_x\) is a Poisson random variable with parameter \(E_x^c\mu\). But then \(\Pr[D_x>n]>0\). That is, an impossible event has non-zero probability, even if it is negligibly small. What is going on?

Physicists are ever alert to the tiniest difference between…

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Tags: Poisson distribution, survival model

Out of line

(Aug 20, 2013)

Regular readers of this blog will be in no doubt of the advantages of survival models over models for the annual mortality rate, qx. However, what if an analyst wants to stick to the historical actuarial tradition of modelling annualised mortality rates? Figure 1 shows a GLM for qx fitted to some mortality data for a large UK pension scheme.

Figure 1. Observed mortality rates (•) and fitted values (-) using a binomial GLM with default canonical link (logit scale). Source: Own calculations using the mortality experience of a large UK pension scheme for the single calendar year 2009.

Figure 1 shows that the GLM provides a good approximation of the mortality patterns. A check of the deviance residuals (not shown) yields…

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Tags: GLM, linearity, survival model

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