The Annals of Mathematical Statistics

A Note on the Estimation of the Mode

Edward J. Wegman

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Abstract

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.

Article information

Source
Ann. Math. Statist., Volume 42, Number 6 (1971), 1909-1915.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177693056

Digital Object Identifier
doi:10.1214/aoms/1177693056

Mathematical Reviews number (MathSciNet)
MR297072

Zentralblatt MATH identifier
0227.62028

JSTOR
links.jstor.org

Citation

Wegman, Edward J. A Note on the Estimation of the Mode. Ann. Math. Statist. 42 (1971), no. 6, 1909--1915. doi:10.1214/aoms/1177693056. https://projecteuclid.org/euclid.aoms/1177693056


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