Electronic Journal of Statistics

Univariate log-concave density estimation with symmetry or modal constraints

Charles R. Doss and Jon A. Wellner

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We study nonparametric maximum likelihood estimation of a log-concave density function $f_{0}$ which is known to satisfy further constraints, where either (a) the mode $m$ of $f_{0}$ is known, or (b) $f_{0}$ is known to be symmetric about a fixed point $m$. We develop asymptotic theory for both constrained log-concave maximum likelihood estimators (MLE’s), including consistency, global rates of convergence, and local limit distribution theory. In both cases, we find the MLE’s pointwise limit distribution at $m$ (either the known mode or the known center of symmetry) and at a point $x_{0}\ne m$. Software to compute the constrained estimators is available in the R package logcondens.mode.

The symmetry-constrained MLE is particularly useful in contexts of location estimation. The mode-constrained MLE is useful for mode-regression. The mode-constrained MLE can also be used to form a likelihood ratio test for the location of the mode of $f_{0}$. These problems are studied in separate papers. In particular, in a separate paper we show that, under a curvature assumption, the likelihood ratio statistic for the location of the mode can be used for hypothesis tests or confidence intervals that do not depend on either tuning parameters or nuisance parameters.

Article information

Electron. J. Statist., Volume 13, Number 2 (2019), 2391-2461.

Received: October 2018
First available in Project Euclid: 23 July 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Mode consistency convergence rate empirical processes convex optimization log-concave shape constraints symmetric

Creative Commons Attribution 4.0 International License.


Doss, Charles R.; Wellner, Jon A. Univariate log-concave density estimation with symmetry or modal constraints. Electron. J. Statist. 13 (2019), no. 2, 2391--2461. doi:10.1214/19-EJS1574. https://projecteuclid.org/euclid.ejs/1563868824

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