This paper studies a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity, and proposes and studies algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class.Expand

It is argued that the bound presents a benchmark for the risk of any estimator in isotonic regression by proving nonasymptotic local minimax lower bounds and an analogue of the authors' bound for model misspecification where the true $\theta$ is not necessarily nondecreasing.Expand

We consider the problem of nonparametric estimation of a convex regression function $$\phi _0$$ϕ0. We study the risk of the least squares estimator (LSE) under the natural squared error loss. We show… Expand

These results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.Expand

We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given positive integer $r$, the $r$-th order trend filtering estimator is defined as the minimizer… Expand

It is shown that the risk of the entirely monotonic LSE is almost parametric (at most $1/n$ up to logarithmic factors) when the true function is well-approximable by a rectangular piecewise constant entirely Monotone function with not too many constant pieces.Expand

An interesting adaptation property of the LSE is derived which is referred to as variable adaptation -- theLSE performs as well as the oracle estimator when estimating a matrix that is constant along each row/column.Expand

Lower bounds involving f-divergences between the underlying probability measures are proved for the minimax risk in estimation problems and special cases and straightforward corollaries of these bounds include well known inequalities for establishing minimax lower bounds such as Fano's inequality, Pinsker's inequality and inequalities based on global entropy conditions.Expand

A minimax optimal solution to the problem of estimating a compact, convex set from finitely many noisy measurements of its support function is presented, based on appropriate regularizations of the least squares estimator.Expand