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2023 Posterior contraction and testing for multivariate isotonic regression
Kang Wang, Subhashis Ghosal
Author Affiliations +
Electron. J. Statist. 17(1): 798-822 (2023). DOI: 10.1214/23-EJS2115

Abstract

We consider the nonparametric regression problem with multiple predictors and an additive error, where the regression function is assumed to be coordinatewise nondecreasing. We propose a Bayesian approach to make inferences on the multivariate monotone regression function, obtain the posterior contraction rate, and construct a universally consistent Bayesian testing procedure for multivariate monotonicity. To facilitate posterior analysis, we temporarily set aside the shape restrictions, and endow a prior on blockwise constant regression functions with independently normally distributed heights. The unknown variance of the error term is either estimated by the marginal maximum likelihood estimate, or is equipped with an inverse-gamma prior. Then the unrestricted block-heights are a posteriori also independently normally distributed given the error variance, by conjugacy. To comply with the shape restrictions, we project samples from the unrestricted posterior onto the class of multivariate monotone functions, inducing the “projection-posterior distribution”, to be used for making an inference. Under an L1-metric, we show that the projection-posterior based on n independent samples contracts around the true monotone regression function at the optimal rate n1(2+d). Then we construct a Bayesian test for multivariate monotonicity based on the posterior probability of a shrinking neighborhood of the class of multivariate monotone functions. We show that the test is universally consistent, that is, the level of the Bayesian test goes to zero, and the power at any fixed alternative goes to one. Moreover, we show that for a smooth alternative function, power goes to one as long as its distance to the class of multivariate monotone functions is at least of the order of the estimation error for a smooth function. To the best of our knowledge, no other test for multivariate monotonicity is available in the Bayesian or the frequentist literature.

Funding Statement

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

Acknowledgments

The authors would like to thank two anonymous referees, the Associate Editor and the Editor for their constructive comments that improved the quality of this paper.

Citation

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Kang Wang. Subhashis Ghosal. "Posterior contraction and testing for multivariate isotonic regression." Electron. J. Statist. 17 (1) 798 - 822, 2023. https://doi.org/10.1214/23-EJS2115

Information

Received: 1 June 2022; Published: 2023
First available in Project Euclid: 1 March 2023

MathSciNet: MR4554662
zbMATH: 07662462
Digital Object Identifier: 10.1214/23-EJS2115

Subjects:
Primary: 60G10 , 62F15 , 62G20

Keywords: Bayesian tests for multivariate monotonicity , contraction rate , Multivariate isotonic regression

Vol.17 • No. 1 • 2023
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