### 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) |
---|---|

Pages (from-to) | 1-25 |

Number of pages | 25 |

Journal | Bernoulli |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2019 |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Bernoulli*,

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

**Adaptive risk bounds in unimodal regression.** / Chatterjee, Sabyasachi; Lafferty, John.

Research output: Contribution to journal › Article

*Bernoulli*, vol. 25, no. 1, pp. 1-25. https://doi.org/10.3150/16-BEJ922

}

TY - JOUR

T1 - Adaptive risk bounds in unimodal regression

AU - Chatterjee, Sabyasachi

AU - Lafferty, John

PY - 2019/2/1

Y1 - 2019/2/1

N2 - 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.

AB - 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.

KW - Isotonic regression

KW - Minimax bounds

KW - Shape constrained inference

KW - Unimodal regression

UR - http://www.scopus.com/inward/record.url?scp=85058479885&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85058479885&partnerID=8YFLogxK

U2 - 10.3150/16-BEJ922

DO - 10.3150/16-BEJ922

M3 - Article

VL - 25

SP - 1

EP - 25

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 1

ER -