Predicting future values of a random function

Prediction from time-series data is traditionally accomplished using parametric, or at least structural, methods. For example, after removing trends, arguing that the time-series is an autoregression, and estimating the autoregressive parameters, we may predict future expected values, conditional on the past. In this paper, motivated by long meteorological maximum-temperature time-series, we suggest alternative approaches founded on functional data analysis. The new techniques make relatively few assumptions about the nature of the data, and allow consistent inference in cases where conventional models are inappropriate. They have both parametric and nonparametric forms. In the former context, our techniques are based on dimension-reduction and least-squares arguments; in the latter, they are founded on distance-based methods and statistical smoothing. We illustrate our method by application to Australian meteorological data, and by a simulation study designed to reflect those data. Theoretical arguments are used to demonstrate statistical consistency.