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June, 1965 A Nonparametric Estimate of a Multivariate Density Function
D. O. Loftsgaarden, C. P. Quesenberry
Ann. Math. Statist. 36(3): 1049-1051 (June, 1965). DOI: 10.1214/aoms/1177700079


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.


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D. O. Loftsgaarden. C. P. Quesenberry. "A Nonparametric Estimate of a Multivariate Density Function." Ann. Math. Statist. 36 (3) 1049 - 1051, June, 1965.


Published: June, 1965
First available in Project Euclid: 27 April 2007

zbMATH: 0132.38905
MathSciNet: MR176567
Digital Object Identifier: 10.1214/aoms/1177700079

Rights: Copyright © 1965 Institute of Mathematical Statistics

Vol.36 • No. 3 • June, 1965
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