Nonparametric estimation of multivariate density and nonparametric regression

Abstract

ABSTRACT The p.d.f.s typically used in hydrology for determination of exceedance probability (e.g. design floods) are typically based on the parametric approach. In the two-or three-dimensional cases and in the case of regression problems the multivariate normal distribution is in common use. Nonparametric density estimators in multivariate random variables are a new approach to estimation and regression. As an alternative to the standard parametric estimators, the nonparametric multivariate Parzen estimator has been used in the analysis. The results of the analysis indicate that the parametric and nonparametric estimators are performing comparatively well. Some conclusions are offered concerning the applications of the nonparametric approaches. INTRODUCTION Various probability distributions are used in hydrology for determination of exceedance probability. Flood frequency analysis is an example, where typically only one-dimensional random variables are considered (Flood Frequency and Risk Analyses, 1986; Kaczmarek, 1970). Sometimes models involving two-or three-dimensional random variables are investigated. For example, multivariate models for low or high water stages were developed by Zielinska (1963, 1964), Yevjevich (1967) and Strupczewski (1967), under the assumption of normality of the underlying probability distribution. Another parametric estimation approach in hydrology is a classical regression problem also based on multivariable normal distribution (Kaczmarek, 1970). Recently investigations based on the nonparametric approach (nonparametric method of estimation (NME)) have been initiated in hydrology (Adamowski, 1985, Feluch, 1987, Adamowski & Feluch, 1988, Schuster & Yakowitz, 1985).