Choosing smoothing parameters in nonparametric curve fitting using kernel contrasts

The choice of smoothing parameter is the central issue in the applications of all types of nonparametric density estimation. There are many smoothing parameter selection methods in kernel density estimation. They are based on the idea of minimizing either an unbiased stochastic measure of the corresponding error criterion or its asymptotic approximation.; All these methods have some limitations. In these estimation methods, the underlying unknown density {dollar}f(x){dollar} is always needed. Since {dollar}f(x){dollar} is unknown, people tried to use various methods to approximate it in order to get the smoothing parameter. Therefore, we are faced with a "catch 22" situation. Some of these methods are subject to a great deal of sample to sample variability and sometimes the method may fail all together for small samples. Other methods depend on some subjective assumptions for starting. In this sense, they are not completely data-driven. Some of these selected smoothing parameters are not even consistent with the theoretical optimal choice.; Our goal is to avoid the above problems by proposing two new methods which are automatic and completely data-driven: the Kernel Contrast Method and the Adjusted Kernel Contrast Method with Parametric Start.; The breakthrough of the kernel contrast estimation method consists of two parts. The first is the introduction of the contrast concept into the error criterion. The second is the recovery of the asymptotic equivalence between the data based choice of {dollar}h, Lambda hsb{lcub}con{rcub}{dollar} and the theoretically optimal choice {dollar} hsb{lcub}opt{rcub}.{dollar} Asymptotic properties of the selected smoothing parameter, {dollar}Lambda hsb{lcub}con{rcub}{dollar} and the constructed density estimator {dollar}tilde f(x){dollar} are discussed. The Monte Carlo studies and the study of a real data example show the nice finite sample behavior of our kernel contrast estimator and how well it performs in practice. We extend our results to the multivariate cases where product kernels are used with a constraint {dollar}hsb1=hsb2=...=hsb{lcub}d{rcub}=h.{dollar}; The adjusted kernel contrast method with parametric start uses a "parametric start," {dollar}fsb0(x; theta),{dollar} where the functional form of this density is assumed known. The adjusted contrast kernel density estimator with parametric start (ps) can be better than the traditional one in the sense of the bias and the mean integrated square error when the parametric start is chosen properly. Asymptotic properties of the selected smoothing parameter, {dollar}Lambda hsb{lcub}conps{rcub}{dollar} and the constructed density estimator {dollar}tilde f(x; fsb0){dollar} are discussed. The Monte Carlo studies show the nice finite sample behavior of our adjusted kernel contrast estimator with parametric start.