### Abstract

We study the statistical properties of the least squares estimator in unimodal sequence estimation. Although closely related to isotonic regression, unimodal regression has not been as extensively studied. We show that the unimodal least squares estimator is adaptive in the sense that the risk scales as a function of the number of values in the true underlying sequence. Such adaptivity properties have been shown for isotonic regression by Chatterjee et al. (Ann. Statist. 43 (2015) 1774–1800) and Bellec (Sharp oracle inequalities for Least Squares estimators in shape restricted regression (2016)). A technical complication in unimodal regression is the non-convexity of the underlying parameter space. We develop a general variational representation of the risk that holds whenever the parameter space can be expressed as a finite union of convex sets, using techniques that may be of interest in other settings.

Original language | English (US) |
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Pages (from-to) | 1-25 |

Number of pages | 25 |

Journal | Bernoulli |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2019 |

### Keywords

- Isotonic regression
- Minimax bounds
- Shape constrained inference
- Unimodal regression

### ASJC Scopus subject areas

- Statistics and Probability

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## Cite this

*Bernoulli*,

*25*(1), 1-25. https://doi.org/10.3150/16-BEJ922