A Consistent Nonparametric Multivariate Density Estimator Based on Statistically Equivalent Blocks

Abstract

Let $x_1, x_2, \cdots, x_m$ be a random sample from a $p$-dimensional random variable $X = (X_1, X_2, \cdots, X_p)$ with probability distribution $P$. It is assumed that $P$ is absolutely continuous with respect to Lebesgue measure, and that the corresponding probability density function is denoted by $f$. If $z = (z_1, z_2, \cdots, z_p)$ is a point at which $f$ is both continuous and positive, an estimator for $f(z)$ based on statistically equivalent blocks is suggested and its consistency is shown. This estimator grew out of work on the nonparametric discrimination problem. Fix and Hodges [2] showed how density estimation could be used in this problem and demonstrated a consistent estimator at points such as $z$. Loftsgaarden and Quesenberry [4] proposed another estimator which is consistent at points such as $z$; their estimator was based on statistically equivalent blocks. Although this estimator is easier to use in practice than that suggested by Fix and Hodges, it does require separate calculations if the sample is to be used to estimate the density at two or more points, and gives complex regions on which the estimate is constant if it is desired to estimate $f$ on some subset of the entire space. The estimator suggested in this paper is consistent at all points at which the two above estimators are consistent and allows the investigator to estimate the density at every point of $p$-dimensional Euclidean space from one construction, as well as providing rectangular regions on which the estimate is constant.