Summary.We consider an application in electricity grid load prediction, where generalized additive models are appropriate, but where the data set's size can make their use practically intractable with existing methods. We therefore develop practical generalized additive model fitting methods for large data sets in the case in which the smooth terms in the model are represented by using penalized regression splines. The methods use iterative update schemes to obtain factors of the model matrix while requiring only subblocks of the model matrix to be computed at any one time.We show that efficient smoothing parameter estimation can be carried out in a well-justified manner. The grid load prediction problem requires updates of the model fit, as new data become available, and some means for dealing with residual auto-correlation in grid load. Methods are provided for these problems and parallel implementation is covered. The methods allow estimation of generalized additive models for large data sets by using modest computer hardware, and the grid load prediction problem illustrates the utility of reduced rank spline smoothing methods for dealing with complex modelling problems.

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We consider opportunity-based age replacement using nonparametric predictive inference (NPI) for the time to failure of a future unit. Based on n observed failure times, NPI provides lower and upper bounds for the survival function for the time to failure X n+1 of a future unit, which lead to upper and lower cost functions, respectively, for opportunitybased age replacement based on the renewal reward theorem. Optimal opportunity-based age replacement strategies for unit n+1 follow by minimising these cost functions. Following this strategy, unit n + 1 is correctively replaced upon failure, or preventively replaced upon the first opportunity after the optimal opportunity-based age replacement threshold. We study the effect of this replacement information for unit n + 1 on the optimal opportunitybased age replacement strategy for unit n + 2. We illustrate our method with examples and a simulation study.Our method is fully adaptive to available data, providing an alternative to the classical approach where the probability distribution of a unit's time to failure is assumed to be known. We discuss the possible use of our method and compare it with the classical approach, where we conclude that in most situations our adaptive method performs very well, but that counter-intuitive results can occur.

We consider the adjustment, based upon a sample of size n, of collections of vectors drawn from either an infinite or finite population. The vectors may be judged to be either Normally distributed or, more generally, second-order exchangeable. We develop the work of Goldstein and Wooff (1998) to show how the familiar univariate finite population corrections (fpc) naturally generalise to individual quantities in the multivariate population. The types of information we gain by sampling are identified with the orthogonal canonical variable directions derived from a generalised eigenvalue problem. These canonical directions share the same coordinate representation for all sample sizes and, for equally defined individuals, all population sizes enabling simple comparisons between both the effects of different sample sizes and of different population sizes. We conclude by considering how the fpc is modified for multivariate cluster sampling with exchangeable clusters. In univariate two-stage cluster sampling we may decompose the variance of the population mean into the sum of the variance of cluster means and the variance of the cluster members within clusters. The first term has a fpc relating to the sampling fraction of clusters, the second term has a fpc relating to the sampling fraction of cluster size. We illustrate how this generalises in the multivariate case. We decompose the variance into two terms: the first relating to multivariate finite population sampling of clusters and the second to multivariate finite population sampling within clusters. We solve two generalised eigenvalue problems to show how to generalise the univariate to the multivariate: each of the two fpcs attaches to one, and only one, of the two eigenbases.

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