Stephen Richards
Articles written by Stephen Richards
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:
\[\tilde{\chi}^2 = \sum_{i=1}^n r_i^2\]
Some points for integration
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.
(Mis-)Estimation of mortality risk
Working with constraints
The name of the game
A chill wind
In a previous blogs I have looked at seasonal fluctuations in mortality, usually with lower mortality in summer and higher mortality in winter. The subject of excess winter deaths is back in the news, as the UK experienced heavy mortality in the winter of 2014/15, as demonstrated in Figure 1.
What — and when — is a 1:200 event?
Reviewing forecasts
Can I interest you in a guaranteed loss-making investment?
Conditional tail expectations
In a recent posting I looked at the calculation of percentiles and quantiles, which underpin many calculations for ICA and Solvency II. Simply put, an \(\alpha\)-quantile is the value which is not expected to be exceeded \(\alpha\times 100\)% of the time. This value is denoted \(Q_{\alpha}\). Mathematically, for a continuous random variable, \(X\), and a given probability level \(\alpha\) we have:
$$\Pr(X\leq Q_\alpha)=\alpha$$