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The paper introduces a semiparametric model for functional data. The warping functions are assumed to be linear combinations of "q" common components, which are estimated from the data (hence the name 'self-modelling'). Even small values of "q" provide remarkable model flexibility, comparable with nonparametric methods. At the same time, this approach avoids overfitting because the common components are estimated combining data across individuals. As a convenient by-product, component scores are often interpretable and can be used for statistical inference (an example of classification based on scores is given). Copyright 2004 Royal Statistical Society.

SA random sample of curves can be usually thought of as noisy realisations of a compound stochastic process X(t)=Z{W (t)}, where Z(t) produces random amplitude variation and W (t) produces random dynamic or phase variation. In most applications it is more important to estimate the so-called structural mean m(t)=E{Z(t)} than the crosssectional mean E{X(t)}, but this estimation problem is difficult because the process Z(t) is not directly observable. In this paper we propose a nonparametric maximum likelihood estimator of m(t). This estimator is shown to be √n-consistent and asymptotically normal under the assumed model and robust to model misspecification. Simulations and a realdata example show that the proposed estimator is competitive with landmark registration, often considered the benchmark, and has the advantage of avoiding time-consuming and often infeasible individual landmark identification.

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