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On risk bounds in isotonic and other shape restricted regression problems
We consider the problem of estimating an unknown $\theta\in {\mathbb{R}}^n$ from noisy observations under the constraint that $\theta$ belongs to certain convex polyhedral cones in ${\mathbb{R}}^n$.Expand
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Stochastically Transitive Models for Pairwise Comparisons: Statistical and Computational Issues
TLDR
In this paper, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. Expand
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Global risk bounds and adaptation in univariate convex regression
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 showExpand
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On matrix estimation under monotonicity constraints
We consider the problem of estimating an unknown $n_1 \times n_2$ matrix $\mathbf{\theta^*}$ from noisy observations under the constraint that $\mathbf{\theta}^*$ is nondecreasing in both rows andExpand
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Covering Numbers for Convex Functions
TLDR
In this paper, we study the covering numbers of the space of convex and uniformly bounded functions in multidimension. Expand
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Adaptive Risk Bounds in Univariate Total Variation Denoising and Trend Filtering
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 minimizerExpand
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Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation
We consider the problem of nonparametric regression when the covariate is $d$-dimensional, where $d \geq 1$. In this paper we introduce and study two nonparametric least squares estimators (LSEs) inExpand
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Lower Bounds for the Minimax Risk Using $f$-Divergences, and Applications
TLDR
Lower bounds involving f-divergences between the underlying probability measures are proved for the minimax risk in estimation problems. Expand
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Spatial Adaptation in Trend Filtering
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Concentration of the spectral measure of large Wishart matrices with dependent entries
We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, butExpand
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