## Abstract

For nonparametric univariate regression under a monotonicity constraint on the regression function f, we study the coverage of a Bayesian credible interval for f(x_{0}), where x_{0} is an interior point. Analysis of the posterior becomes a lot more tractable by considering a “projection-posterior” distribution based on a finite random series of step functions with normal basis coefficients as a prior for f. A sample f from the resulting conjugate posterior distribution is projected on the space of monotone increasing functions to obtain a monotone function f^{∗} closest to f, inducing the “projection-posterior.” We use projection-posterior samples to obtain credible intervals for f(x_{0}). We obtain the asymptotic coverage of the credible interval thus constructed and observe that it is free of nuisance parameters involving the true function. We observe a very interesting phenomenon that the coverage is typically higher than the nominal credibility level, the opposite of a phenomenon observed by Cox (Ann. Statist. 21 (1993) 903-923) in the Gaussian sequence model. We further show that a recalibration gives the right asymptotic coverage by starting from a lower credibility level that can be explicitly calculated.

Original language | English (US) |
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Pages (from-to) | 1011-1028 |

Number of pages | 18 |

Journal | Annals of Statistics |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - 2021 |

Externally published | Yes |

## Keywords

- Chernoff's distribution
- Coverage
- Credible set
- Monotonicity
- Nonparametric regression
- Projection-posterior

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty