Nonparametric maximum likelihood estimation of the structural mean of a sample of curves

SUMMARY 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 µ(t) = E{Z(t)} than the cross-sectional mean E{X(t)}, but this estimation problem is difficult because the process Z(t) is not directly observable. In this article we propose a nonparametric maximum likelihood estimator of µ(t). This es-timator is shown to be √ n-consistent and asymptotically normal under the model assumed and robust to model misspecification. Simulations and a real-data 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.