Nonparametric binary discrimination. Methods for estimating the smoothing para. (Discrimination binaire non paramétrique. Méthodes d’estimation du paramètre de lissage.)

*(French)* Zbl 0972.62510
Summary: The kernel method for estimating the cell probabilities of a multivariate discrete distribution, due to Aitchison and Aitken (1976), depends crucially on an unknown smoothing parameter $\lambda $. Most of the methods for choosing the smoothing parameter are discussed in the context of density estimation. The choice may be based on a pseudo-likelihood or on loss functions for the estimation of the density. In this setting, we show how to apply resampling methods (cross-validation and bootstrap) to estimating the smoothing parameters. If the main interest is not in density estimation but in discrimination, alternative methods for choosing $\lambda $ from the discrimination viewpoint may yield better performance for separation of groups. Methods of this type have been proposed by Tutz (1986, 1989) for discrete kernels and more recently by Hall and Wand (1988). In the same setting, we propose a method, estimating $\lambda $ explicitly, based on minimization of the leaving-one-out estimator of the error rate, without using the iterative method. Moreover, we extend the method of bootstrap to Hall and Wand’s approach in the case of two groups. An example is given to illustrate the practical behaviour of all these methods.

##### MSC:

62G05 | Nonparametric estimation |

62H30 | Classification and discrimination; cluster analysis (statistics) |