## The Annals of Mathematical Statistics

### A Nonparametric Estimate of a Multivariate Density Function

#### Abstract

Let $x_1, \cdots, x_n$ be independent observations on a $p$-dimensional random variable $X = (X_1, \cdots, X_p)$ with absolutely continuous distribution function $F(x_1, \cdots, x_p)$. An observation $x_i$ on $X$ is $x_i = (x_{1i}, \cdots, x_{pi})$. The problem considered here is the estimation of the probability density function $f(x_1, \cdots, x_p)$ at a point $z = (z_1, \cdots, z_p)$ where $f$ is positive and continuous. An estimator is proposed and consistency is shown. The problem of estimating a probability density function has only recently begun to receive attention in the literature. Several authors [Rosenblatt (1956), Whittle (1958), Parzen (1962), and Watson and Leadbetter (1963)] have considered estimating a univariate density function. In addition, Fix and Hodges (1951) were concerned with density estimation in connection with nonparametric discrimination. Cacoullos (1964) generalized Parzen's work to the multivariate case. The work in this paper arose out of work on the nonparametric discrimination problem.

#### Article information

Source
Ann. Math. Statist. Volume 36, Number 3 (1965), 1049-1051.

Dates
First available in Project Euclid: 27 April 2007

http://projecteuclid.org/euclid.aoms/1177700079

Digital Object Identifier
doi:10.1214/aoms/1177700079

Mathematical Reviews number (MathSciNet)
MR176567

Zentralblatt MATH identifier
0132.38905

JSTOR