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Adaptive confidence sets in shape restricted regression

Banff International Research Station for Mathematical Innovation and Discovery

We construct adaptive confidence sets in isotonic and convex regression. In univariate isotonic regression, if the true parameter is piecewise constant with $k$ pieces, then the Least-Squares estimator achieves a parametric rate of order $k/n$ up to logarithmic factors. We construct honest confidence sets that adapt to the unknown number of pieces of the true parameter. The proposed confidence set enjoys uniform coverage over all non-decreasing functions. Furthermore, the squared diameter of the confidence set is of order $k/n$ up to logarithmic factors, which is optimal in a minimax sense. In univariate convex regression, we construct a confidence set that enjoys uniform coverage and such that its diameter is of order $q/n$ up to logarithmic factors, where $q-1$ is the number of changes of slope of the true regression function. We will also discuss application of the presented techniques to sparse linear regression.

Mathematics; Statistics; Operations research, mathematical programming

video/mp4

Author affiliation: Rutgers

2018-07-31

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/66625

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/