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A NONPARAMETRIC ESTIMATE OF A MULTIVARIATE DENSITY FUNCTION

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Article Contentsp.1049p.1050p.1051Issue Table of ContentsThe Annals of Mathematical Statistics, Vol. 36, No. 3 (Jun., 1965), pp. 747-1095Front MatterAdmissible Bayes Character of T-, R-, and Other Fully Invariant Tests for Classical Multivariate Normal Problems [pp.747-770]Gaussian Processes on Several Parameters [pp.771-788]Moments of Randomly Stopped Sums [pp.789-799]A Sharper Form of the Borel-Cantelli Lemma and the Strong Law [pp.800-807]On the Convergence of Moments in the Central Limit Theorem [pp.808-818]On the Lifting Property (V) [pp.819-828]Invariant Conditional Distributions [pp.829-846]On Some Robust Estimates of Location [pp.847-858]Bounds on the Maximum Sample Size of a Bayes Sequential Procedure [pp.859-878]Sequential Compound Estimators [pp.879-889]On Information in Statistics [pp.890-896]Order Statistics and Statistics of Structure (d) [pp.897-906]Exact Moments and Percentage Points of the Order Statistics and the Distribution of the Range From the Logistic Distribution [pp.907-920]Quantiles and Medians [pp.921-925]Some Basic Properties of the Incomplete Gamma Function Ratio [pp.926-937]Properties of the Extended Hypergeometric Distribution [pp.938-945]Renewal Theory in the Plane [pp.946-955]Bernard Friedman's Urn [pp.956-970]Spectral Analysis with Randomly Missed Observations: The Binomial Case [pp.971-977]Asymptotic Inference in Markov Processes [pp.978-992]Extreme Values in Uniformly Mixing Stationary Stochastic Processes [pp.993-999]On the Asymptotic Power of the One-Sample Kolmogorov-Smirnov Tests [pp.1000-1018]Several k-Sample Kolmogorov-Smirnov Tests [pp.1019-1026]Estimation of Probability Density [pp.1027-1031]Estimation of Jumps, Reliability and Hazard Rate [pp.1032-1040]NotesA Limit Theorem for Sums of Minima of Stochastic Variables [pp.1041-1042]The Expected Number of Zeros of a Stationary Gaussian Process [pp.1043-1046]Density Estimation in a Topological Group [pp.1047-1048]A Nonparametric Estimate of a Multivariate Density Function [pp.1049-1051]A Note on Midrange [pp.1052-1054]Some Bounds for Expected Values of Order Statistics [pp.1055-1057]Note on the Wilcoxon-Mann-Whitney Statistic [pp.1058-1060]On the Likelihood Ratio Test of a Normal Multivariate Testing Problem II [pp.1061-1065]Factorial Distributions [pp.1066-1068]Correction Notes: Correction to "Some Rényi Type Limit Theorems for Empirical Distribution Functions" [p.1069]Correction Notes: Correction to "On the Equivalence of Polykays of the Second Degree and Σ'S" [p.1069]Correction Notes: Correction to Abstract "Tables for the Distribution of the Maximum of Correlated Chi-Square Variates with One Degree of Freedom" [p.1069]Book Reviewuntitled [pp.1070-1072]Abstracts of Papers [pp.1073-1087]News and Notices [pp.1088-1094]Publications Received [p.1095]Back MatterA Nonparametric Estimate of a Multivariate Density FunctionAuthor(s): D. O. Loftsgaarden and C. P. QuesenberrySource: The Annals of Mathematical Statistics, Vol. 36, No. 3 (Jun., 1965), pp. 1049-1051Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2238216Accessed: 04/03/2009 09:53Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/action/showPublisher?publisherCode=ims.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with thescholarly community to preserve their work and the materials they rely upon, and to build a common research platform thatpromotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected] of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to TheAnnals of Mathematical Statistics.http://www.jstor.orgA NONPARAMETRIC ESTIMATE OF A MULTIVARIATE DENSITY FUNCTION' BY D. 0. LOFTSGAARDEN AND C. P. - QUESENBERRY Montana State College 1. Introduction and summary. Let xi, * **, xn be independent observations on a p-dimensional random variable X = (X1, , X,) with absolutely con- tinuous distribution function F(x1, ***, x,). An observation xi on X is xi = (xii, * * *, xpi). The problem considered here is the estimation of the probability density function f(x1, * * *, xp) at a point z = (z,, * - *, zp) 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 con- sidered 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 discrimina- tion problem. 2. Preliminaries and notation. Let d(x, z) represent the p-dimensional Euclid- ean distance function Ix - z. If z is the point given in the first paragraph of the preceding section, a p-dimensional hypersphere about z of radius r will be designated by Sr,z, i.e. Sr,z = {x I d(x, z) < r}. The volume or measure of the hypersphere Sr,z will be called Ar,z,. Ar,z is equal to 2rp7rp/2/pr(p/2). Using this notation and noting that Ar,z 0 if and only if r O-0, we have (2.1) f(z1, - ,zp,) = limr,o P(Sr,z)/Ar,z, i.e. there exists an R such that if r < R then (2.2) P(Sr,z)/Ar,z - f(z X * * , zP) I < Ec for arbitrary e > 0. 3. A consistent density function estimator. According to (2.2), P(Sr,z)/Ar,z can be made as near f(z1 , * * *, zp) as one chooses by letting r approach zero. P(Sr,z) is unknown since it depends on the density f being estimated. The approach used here is to find a good estimate for


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