### Abstract

We study nonparametric maximum likelihood estimation of a log-concave density function f0 which is known to satisfy further constraints, where either (a) the mode m of f0 is known, or (b) f0 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 x0≠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 f0. 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.

Original language | English (US) |
---|---|

Pages (from-to) | 2391-2461 |

Number of pages | 71 |

Journal | Electronic Journal of Statistics |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2019 |

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### Keywords

- Consistency
- Convergence rate
- Convex optimization
- Empirical processes
- Log-concave
- Mode
- Shape constraints
- Symmetric

### Cite this

*Electronic Journal of Statistics*,

*13*(2), 2391-2461. https://doi.org/10.1214/19-EJS1574

**Univariate log-concave density estimation with symmetry or modal constraints.** / Doss, Charles R; Wellner, Jon A.

Research output: Contribution to journal › Article

*Electronic Journal of Statistics*, vol. 13, no. 2, pp. 2391-2461. https://doi.org/10.1214/19-EJS1574

}

TY - JOUR

T1 - Univariate log-concave density estimation with symmetry or modal constraints

AU - Doss, Charles R

AU - Wellner, Jon A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We study nonparametric maximum likelihood estimation of a log-concave density function f0 which is known to satisfy further constraints, where either (a) the mode m of f0 is known, or (b) f0 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 x0≠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 f0. 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.

AB - We study nonparametric maximum likelihood estimation of a log-concave density function f0 which is known to satisfy further constraints, where either (a) the mode m of f0 is known, or (b) f0 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 x0≠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 f0. 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.

KW - Consistency

KW - Convergence rate

KW - Convex optimization

KW - Empirical processes

KW - Log-concave

KW - Mode

KW - Shape constraints

KW - Symmetric

UR - http://www.scopus.com/inward/record.url?scp=85070969417&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85070969417&partnerID=8YFLogxK

U2 - 10.1214/19-EJS1574

DO - 10.1214/19-EJS1574

M3 - Article

AN - SCOPUS:85070969417

VL - 13

SP - 2391

EP - 2461

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

SN - 1935-7524

IS - 2

ER -