The prediction of future mortality rates is a problem of fundamental importance for the insurance and pensions industry. We show how the method of P-splines can be extended to the smoothing and forecasting of two-dimensional mortality tables. We use a penalized generalized linear model with Poisson errors and show how to construct regression and penalty matrices appropriate for two-dimensional modelling. An important feature of our method is that forecasting is a natural consequence of the smoothing process. We illustrate our methods with two data sets provided by the Continuous Mortality Investigation Bureau, a central body for the collection and processing of UK insurance and pensions data.

Data with an array structure are common in statistics, and the design or regression matrix for analysis of such data can often be written as a Kronecker product. Factorial designs, contingency tables and smoothing of data on multidimensional grids are three such general classes of data and models. In such a setting, we develop an arithmetic of arrays which allows us to define the expectation of the data array as a sequence of nested matrix operations on a coefficient array. We show how this arithmetic leads to low storage, high speed computation in the scoring algorithm of the generalized linear model. We refer to a generalized linear array model and apply the methodology to the smoothing of multidimensional arrays. We illustrate our procedure with the analysis of three data sets: mortality data indexed by age at death and year of death, spatially varying microarray background data and disease incidence data indexed by age at death, year of death and month of death. Copyright 2006 Royal Statistical Society.

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We consider the application of P-splines (Eilers and Marx, 1996) to three classes of models with smooth components: semiparametric models, models with serially correlated errors, and models with heteroscedastic errors. We show that P-splines provide a common approach to these problems. We set out a simple nonparametric strategy for the choice of the P-spline parameters (the number of knots, the degree of the P-spline, and the order of the penalty) and use mixed model (REML) methods for smoothing parameter selection. We give an example of a model in each of the three classes and analyse appropriate data sets.

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