Nonparametric Regression Curves Research Articles

SEMIPARAMETRIC COMPARISON OF REGRESSION CURVES VIA NORMAL LIKELIHOODS

Summary Hardle & 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 Hardle & Marron, and so the latter technique is in fact optimal in this sense. We discuss the implications of these conclusions for the aperiodic case.

Discussion of the Paper by Hall and Johnstone

Discussion of the Paper by Hall and Johnstone

Semiparametric Comparison of Regression Curves

The comparison of nonparametric regression curves is considered. It is assumed that there are parametric (possibly nonlinear) transformations of the axes which map one curve into the other. Estimation and testing of the parameters in the transformations are studied. The rate of convergence is $n^{-1/2}$ although the nonparametric components of the model typically have a rate slower than that. A statistic is provided for testing the validity of a given completely parametric model.

Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting

SUMMARY Non-parametric regression using cubic splines is an attractive, flexible and widely-applicable approach to curve estimation. Although the basic idea was formulated many years ago, the method is not as widely known or adopted as perhaps it should be. The topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology. Particular subjects covered include the basic principles of the method; the relation with moving average and other smoothing methods; the automatic choice of the amount of smoothing; and the use of residuals for diagnostic checking and model adaptation. The question of providing inference regions for curves – and for relevant properties of curves – is approached via a finite-dimensional Bayesian formulation.

A graphical analysis of the interrelationships among waterborne asbestos, digestive system cancer and population density.

Five statistical procedures were used to partial the correlation between waterborne asbestos and digestive site cancer for the putative effects of population density. These include: analysis based on a data subset with roughly homogeneous population density; standard residual analysis (partial correlation); conditional probability integral transformation; analysis based upon ranked data, and use of logarithmic transformation. Nonparametric regression graphical techniques are applied to examine the nature or shape of the asbestos-cancer dose-response curve. Evidence is presented that suggests that there is considerable difference between analyses involving nonhigh-density tracts and non-San Francisco tracts. Evidence is also presented that the modal-type nonparametric regression curve forks or bifurcates when adjustment is made for population density.