## Abstract

We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on Rp and incorporating within it symmetry assumptions, which facilitate scalable estimation algorithms and can mitigate the curse of dimensionality. Our main symmetry assumption is that the super-level sets of the density are K-homothetic (i.e., scalar multiples of a convex body K ⊆ R^{p}). When K is known, we prove that the K-homothetic log-concave maximum likelihood estimator based on n independent observations from such a density achieves the minimax optimal rate of convergence with respect to, for example, squared Hellinger loss, of order n^{−4}/^{5}, independent of p. Moreover, we show that the estimator is adaptive in the sense that if the data generating density admits a special form, then a nearly parametric rate may be attained. We also provide worst case and adaptive risk bounds in cases where K is only known up to a positive definite transformation, and where it is completely unknown and must be estimated nonparametrically. Our estimation algorithms are fast even when n and p are on the order of hundreds of thousands, and we illustrate the strong finite-sample performance of our methods on simulated data.

Original language | English (US) |
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Pages (from-to) | 650-672 |

Number of pages | 23 |

Journal | Annals of Statistics |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - 2021 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Density estimation
- High-dimensional statistics
- Nonparametric estimation
- Shape-constrained estimation