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Activity Number: 411 - Nonparametric Testing
Type: Contributed
Date/Time: Tuesday, July 30, 2019 : 2:00 PM to 3:50 PM
Sponsor: Section on Nonparametric Statistics
Abstract #305363
Title: Optimal Confidence Bands Under Shape Restriction in Multidimension
Author(s): Pratyay Datta* and Bodhisattva Sen
Companies: Columbia University ASA Student Chapter and Columbia University
Keywords: Confidence Bands ; Adaptivity ; Holder Class of functions; Additive model; Shape constrained functions; Regression

In this Paper we propose to find confidence bands for regression function in multi-dimension. For simplicity we look at standard continuous white noise regression model in the d-dimension hypercube. We use the multidimensional multiscale statistic (Which is an extension of the multiscale statistic proposed by D\"umbgen(2001)) to construct confidence bands for many shape restricted classes e.g. multivariate isotonic, multivariate convex etc. Our confidence bands are shown to have guaranteed finite-sample coverage probability. Our proposed confidence bands are also shown to be adaptive and optimal (in an appropriate sense) with respect to the smoothness of the underlying regression function. Moreover, these bands are adaptive to the intrinsic dimension (as opposed to the ambient dimension d) of the regression function. We have also constructed asymptotic confidence bands for the "completely monotone" (functions that have a density with respect to the Lebesgue measure) functions and shown the rate of convergence for our confidence band to be dimension free. We have also done the same under the highly useful additive model constraints.

Authors who are presenting talks have a * after their name.

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