## Mortality by the book

Our book, *Modelling Mortality with Actuarial Applications*, will appear in Spring 2018. I wrote the second of the three parts, where I describe the modelling and forecasting of aggregate mortality data, such as provided by the Office for National Statistics, the Human Mortality Database or indeed by any insurer whose own data is suitable. I have divided my contribution into four chapters. In the first chapter I deal with one-dimensional data, for example, deaths by age for a given year. The Gompertz model is used to introduce the regression-based approach; estimation is initially by least squares, but by the end of the chapter I use generalized linear models (GLMs) with both Poisson and binomial errors.

An important feature of all four chapters is the use of the statistical language R. Models in all four chapters are fitted with R's powerful model-fitting commands and emphasis is on model definition and interpretation of output. Sample code is given in the text and there will be an online repository where the full code for the majority of the examples in my chapters is provided.

In my second chapter I stay in one dimension and introduce smoothing methods. I start with Whittaker's ground-breaking smoothing method; this provides a nice introduction to the more general method of P-splines. There is a careful discussion of model fit, model dimension, overdispersion and the choice of an appropriate level of smoothing. Models are fitted with the R package MortalitySmooth.

In the third chapter I discuss modelling mortality by age and year. The Lee-Carter (LC) model, the Cairns-Blake-Dowd (CBD) model and the two-dimensional P-spline model are described. Again, model fitting is straightforward for all three models with R's high-level functions.

In my final chapter I discuss forecasting with the three models discussed in the previous chapter. Autoregressive integrated moving-average (ARIMA) models are described. For the LC and CBD models forecasting is done with R's arima() function, while forecasting in two-dimensional P-spline models uses penalty forecasting. In all three cases I lay particular emphasis on the accuracy of the forecast. The measurement of both parameter and stochastic error is described and illustrated, and my contribution finishes with a discussion of model risk.

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