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Description

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.