SEMIPARAMETRIC COMPARISON OF REGRESSION CURVES VIA NORMAL LIKELIHOODS
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Hrdle & Marron (1990) treated the problem of semiparametric comparison of nonparametric regression curves by proposing a kernelbased estimator derived by minimizing a version of weighted integrated squared error. The resulting estimators of unknown transformation parameters are nconsistent, 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 does not depend at all on the manner of estimating error variances, provided they are estimated nconsistently. The optimal kernelbased estimator derived via these considerations is asymptotically equivalent to a periodic version of that suggested by Hrdle & Marron, and so the latter technique is in fact optimal in this sense. We discuss the implications of these conclusions for the aperiodic case. Copyright 1992, Wiley Blackwell. All rights reserved