Non-Parametric Regression in Curve Fitting

Abstract

In the present paper we consider a number of non-parametric regression methods for smoothing curves. These comprise (i) series estimators (classical Fourier and polynomial), (ii) cubic smoothing splines and (iii) least-squares splines. The methods discussed in this paper are intended to promote the understanding and extend the practicability of the non-parametric smoothing methodology. Particular emphasis was placed upon the ties between these methods and polynomial regression. The generalized cross-validation criterion was used to select the optimum value of smoothing parameters. The classical Fourier series estimators led to poor fit with the largest variance. In view of the ill-conditioned correlated design matrix, the B-spline coefficients were used to fit the least-squares splines. If the remainder term of the regression function is too small, the smoothing and least- squares splines result in a close fit to the polynomial regression estimators. The methods are illustrated with real- life data, i.e. the ratios of weight to height of Kuwaiti pre-school-age boys collected in a national survey. Other relevant techniques (e.g. smoothed isotonic regression, non-parametric Bayesian regression and local smoothing) are also discussed.