SEMIPARAMETRIC COMPARISON OF REGRESSION CURVES VIA NORMAL LIKELIHOODS

Härdle & Marron (1990) treated the problem of semiparametric comparison of nonparametric regression curves by proposing a kernel-based estimator derived by minimizing a version of weighted integrated squared error. The resulting estimators of unknown transformation parameters are n-consistent, which prompts a consideration of issues. of optimality. We show that when the unknown mean function is periodic, an optimal nonparametric estimator may be motivated by an elegantly simple argument based on maximum likelihood estimation in a parametric model with normal errors. Strikingly, the asymptotic variance of an optimal estimator of θ does not depend at all on the manner of estimating error variances, provided they are estimated n-consistently. The optimal kernel-based estimator derived via these considerations is asymptotically equivalent to a periodic version of that suggested by Härdle & Marron, and so the latter technique is in fact optimal in this sense. We discuss the implications of these conclusions for the aperiodic case.