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December, 1971 A Note on the Estimation of the Mode
Edward J. Wegman
Ann. Math. Statist. 42(6): 1909-1915 (December, 1971). DOI: 10.1214/aoms/1177693056


Let $X_1, \cdots, X_n$ be a sample from a unimodal distribution, $F$, and let $\{a_n\}$ be a sequence converging to zero. A nonparametric estimate of the mode is the center of the interval of length $2a_n$ containing the most observations. This estimate is shown to be strongly consistent and conditions on the speed at which $a_n$ may converge to zero are given. This estimator of the mode is related to the naive density estimator, $(F_n(x + a_n) - F_n(x - a_n))/2a_n$, where $F_n$ is the empirical distribution function. A simple strong consistency result for this naive density estimator is given. Also other estimators of the mode are discussed briefly and an application of estimators of the mode is mentioned.


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Edward J. Wegman. "A Note on the Estimation of the Mode." Ann. Math. Statist. 42 (6) 1909 - 1915, December, 1971.


Published: December, 1971
First available in Project Euclid: 27 April 2007

zbMATH: 0227.62028
MathSciNet: MR297072
Digital Object Identifier: 10.1214/aoms/1177693056

Rights: Copyright © 1971 Institute of Mathematical Statistics


Vol.42 • No. 6 • December, 1971
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