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2021 Convergence rates for Bayesian estimation and testing in monotone regression
Moumita Chakraborty, Subhashis Ghosal
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Electron. J. Statist. 15(1): 3478-3503 (2021). DOI: 10.1214/21-EJS1861

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 n13 under the L1-metric and at a nearly optimal rate under the empirical Lp-metrics for 0<p2. 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.

Funding Statement

Research is partially supported by NSF grant number DMS-1916419.

Citation

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Moumita Chakraborty. Subhashis Ghosal. "Convergence rates for Bayesian estimation and testing in monotone regression." Electron. J. Statist. 15 (1) 3478 - 3503, 2021. https://doi.org/10.1214/21-EJS1861

Information

Received: 1 June 2020; Published: 2021
First available in Project Euclid: 22 June 2021

Digital Object Identifier: 10.1214/21-EJS1861

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