## Abstract

Shape restrictions such as monotonicity on functions often arise naturally in statistical modeling. We consider a Bayesian approach to the estimation of a monotone regression function and testing for monotonicity. We construct a prior distribution using piecewise constant functions. For estimation, a prior imposing monotonicity of the heights of these steps is sensible, but the resulting posterior is harder to analyze theoretically. We consider a “projection-posterior” approach, where a conjugate normal prior is used, but the monotonicity constraint is imposed on posterior samples by a projection map onto the space of monotone functions. We show that the resulting posterior contracts at the optimal rate n−^{1/3} under the L_{1}-metric and at a nearly optimal rate under the empirical Lp-metrics for 0 < p ≤ 2. The projection-posterior approach is also computationally more convenient. We also construct a Bayesian test for the hypothesis of monotonicity using the posterior probability of a shrinking neighborhood of the set of monotone functions. We show that the resulting test has a universal consistency property and obtain the separation rate which ensures that the resulting power function approaches one.

Original language | English (US) |
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Pages (from-to) | 3478-3503 |

Number of pages | 26 |

Journal | Electronic Journal of Statistics |

Volume | 15 |

Issue number | 1 |

DOIs | |

State | Published - 2021 |

Externally published | Yes |

## Keywords

- Bayesian testing
- Monotonicity
- Posterior contraction
- Projection-posterior

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty