| Abstract: | The problem of semiparametric comparison of nonparametric regression curves was treated by Härdle & Marron (1990), who proposed a kernel-based estimator derived by minimising a version of weighted integrated squared error. The resulting estimators of unknown transformation parameters are root n-consistent, and that feature prompts 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 [theta] does not depend at all on the manner of estimating error variances, provided they are estimated root 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. |
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