Posterior contraction and testing for multivariate isotonic regression. (English) Zbl 07662462
The aim of this paper is to generalize the testing results of M. Chakraborty and S. Ghosal [Electron. J. Stat. 15, No. 1, 3478–3503 (2021; Zbl 1472.62106)] from unidimensional to multidimensional predictors. Using the projection technique a Bayesian approach to multivariate monotone regression is considered. It is shown that the resulting induced posterior supported on block-wise constant multivariate monotone function contracts at the optimal rate with respect to an \(\mathbb L_1\)-metric. A new \(\mathbb L_1\)-approximation property for multivariate monotonic functions by piecewise constant function is obtained, and a test for multivariate monotonicity based on the posterior probability of a slight enlargement of the set of multivariate monotone functions is constructed. Also it is shown that the resulting Bayesian test is universally consistent in that the size of the test goes to zero, and the power goes to one at any fixed alternative, as the sample size increases to infinity. The paper is organized in five sections and an appendix. After a presentation of the topic in the first section of the paper, in the second section three assumptions are given and the prior distribution and the projection-posterior approach are described. The main results are presented in in the third section, and in the fourth section simulation study is given to verify the proposed estimation and testing procedure. The proofs of the main theorems are given in the fifth section of the paper, and in Appendix some auxiliary results are presented.
Reviewer: Ilie Valuşescu (Bucureşti)
MSC:
| 62G10 | Nonparametric hypothesis testing |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62F15 | Bayesian inference |
Keywords:
multivariate isotonic regression; contraction rate; Bayesian tests for multivariate monotonicityCitations:
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