Multivariate Boundary Kernels and a Continuous Least Squares Principle

Abstract

Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean-squared error or sup-norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non-negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel-generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss±Markov theorem.

Nonparametric density estimation for multivariate bounded data

“…It would be interesting to perform another detailed simulation analysis to investigate alternative bandwidth selection methods (i.e. biased cross validation or bootstrap) and to compare our estimator with the Müller and Stadtmüller (1999) estimator. The results can also be extended to the multivariate time series case, the censored data case or further developed for multivariate nonparametric regression and for multivariate data defined on more involved supports.…”

“…We can, for example, combine beta with gamma kernels if the supports of the variables are the unit interval and the positive real line, respectively. Furthermore, the nonparametric estimator with a gamma or beta kernel is always nonnegative, while the Müller and Stadtmüller (1999) estimator can be negative. The latter estimator also requires an additional bandwidth parameter and a weighting function.…”

“…In the first illustration we reproduce the density estimates of the second example in Müller and Stadtmüller (1999). The data come from the 1970 US draft lottery data and are available on the Statlib website.…”

“…Although the consequences of the boundary problem in multivariate dimensions are much more severe, solutions to the problem are not well investigated. Müller and Stadtmüller (1999) propose boundary kernels for multivariate data defined on arbitrary support by selecting the kernels that minimize a variational problem. In fact, they extend the minimum variance selection principle kernel used to select the optimal kernel in the interior region, as in Epanechnikov (1969) and Granovsky and Müller (1991).…”

Nonparametric estimation of copula functions for dependence modelling

“…(Müller, 1991(Müller, , 1993Fan and Gijbels, 1992;and Jones, 1993) that kernel estimator encounters boundary bias due to a partial loss of kernel weight near the boundaries. An account on kernel estimation with multivariate boundary regions, which is the most relevant to the copula case, is given in Müller and Stadtmüller (1999). As it turns out the estimator of Fermanian and Scailett (2004) is subject to the boundary bias which causes the estimator no longer consistent near all four edges of the unit square.…”