Asymptotic Nonparametric Spline Density Estimation in Several Variables

Abstract

In [8] we have announced a linear spline method for nonparametric density and distribution estimation on the real line. More detailed discussion of the asymptotic results on the real line is presented in [12]. In this paper asymptotic properties of a large family of such estimators in several variables are discussed. It is interesting that the results do not require existence of derivatives of the density in question. For the asymptotic results on distribution functions the knowledge of the behavior of the second modulus of smoothness in the L^∞ norm is sufficient and in the case of density estimation the knowledge of the behavior of the second modulus of smoothness in the L¹ norm and of the tail function is needed. The method gives simultaneously estimators for density and distribution functions. At the end we have derived the explicit window function in terms of a given finite simple sample and of the window parameter. The window function makes possible in each case to determine the size of the optimal window parameter. In obtaining the results the technique of approximation theory, in particular by tensor product splines, is used following the same guide lines as presented in [9] — [12].