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

Abstract

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 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 μ( 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.