## Abstract

We study the adaptation properties of the multivariate log-concave maximum likelihood estimator over three subclasses of log-concave densities. The first consists of densities with polyhedral support whose logarithms are piecewise affine. The complexity of such densities f can be measured in terms of the sum

Γ(f) of the numbers of facets of the subdomains in the polyhedral subdivision of the support induced by f. Given n independent observations from a d-dimensional log-concave density with d ∈ {2,3}, we prove a sharp oracle inequality, which in particular implies that the Kullback–Leibler risk of the log-concave maximum likelihood estimator for such densities is bounded above by

Γ(f)/n, up to a polylogarithmic factor. Thus, the rate can be essentially parametric, even in this multivariate setting. For the second type of adaptation, we consider densities that are bounded away from zero on a polytopal support; we show that up to polylogarithmic factors, the log-concave maximum likelihood estimator attains the rate n−4/7 when d=3, which is faster than the worst-case rate of n–¹⁄². Finally, our third type of subclass consists of densities whose contours are well separated; these new classes are constructed to be affine invariant and turn out to contain a wide variety of densities, including those that satisfy Hölder regularity conditions. Here, we prove another sharp oracle inequality, which reveals in particular that the log-concave maximum likelihood estimator attains a risk bound of order n-min(β+3β+47) when d=3 over the class of β-Hölder log-concave densities with β>1, again up to a polylogarithmic factor.

Γ(f) of the numbers of facets of the subdomains in the polyhedral subdivision of the support induced by f. Given n independent observations from a d-dimensional log-concave density with d ∈ {2,3}, we prove a sharp oracle inequality, which in particular implies that the Kullback–Leibler risk of the log-concave maximum likelihood estimator for such densities is bounded above by

Γ(f)/n, up to a polylogarithmic factor. Thus, the rate can be essentially parametric, even in this multivariate setting. For the second type of adaptation, we consider densities that are bounded away from zero on a polytopal support; we show that up to polylogarithmic factors, the log-concave maximum likelihood estimator attains the rate n−4/7 when d=3, which is faster than the worst-case rate of n–¹⁄². Finally, our third type of subclass consists of densities whose contours are well separated; these new classes are constructed to be affine invariant and turn out to contain a wide variety of densities, including those that satisfy Hölder regularity conditions. Here, we prove another sharp oracle inequality, which reveals in particular that the log-concave maximum likelihood estimator attains a risk bound of order n-min(β+3β+47) when d=3 over the class of β-Hölder log-concave densities with β>1, again up to a polylogarithmic factor.

Original language | English |
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Pages (from-to) | 129-153 |

Number of pages | 25 |

Journal | The Annals of Statistics |

Volume | 49 |

Issue number | 1 |

Early online date | 29 Jan 2021 |

DOIs | |

Publication status | Published - 1 Feb 2021 |