Reduction of Potential Boundary Bias in Kernel Cumulative Distribution Estimation in Univariate and Multivariate Settings

We propose a new method for nonparametric estimation of a probability distribution and its endpoint. As is well known, the kernel distribution estimator suffers from the boundary bias problem when the endpoint is finite. When the support of the density is unknown, it is necessary to estimate it first. Hall and Park (Annals Stat 1460-1479, 2002) proposed estimating the endpoint by using the sample maximum, which is substituted for the endpoint in a density estimator. We propose a new estimator of the endpoint, which is intended to reduce the boundary bias of the distribution estimator. It is demonstrated that the proposed distribution estimator is numerically superior in the sense of an integrated squared error. Moreover, we discuss the extension of the proposed method to a multivariate case. A new method for estimating the joint probability distribution is also free from the boundary bias and performs numerically better than the naive distribution estimator.