Local continuity of log-concave projection, with applications to estimation under model misspecification

Approximation and Estimation of s-Concave Densities via Rényi Divergences

In this paper, we study the approximation and estimation of $s$-concave densities via Rényi divergence. We first show that the approximation of a probability measure $Q$ by an $s$-concave densities exists and is unique via the procedure of minimizing a divergence functional proposed by Koenker and Mizera (2010) if and only if $Q$ admits full-dimensional support and a first moment. We also show continuity of the divergence functional in $Q$: if $Q_n \to Q$ in the Wasserstein metric, then the projected densities converge in weighted $L_1$ metrics and uniformly on closed subsets of the continuity…

Additive Bayesian variable selection under censoring and misspecification

We discuss the role of misspecification and censoring on Bayesian model selection in the contexts of right-censored survival and concave log-likelihood regression. Misspecification includes wrongly assuming the censoring mechanism to be non-informative. Emphasis is placed on additive accelerated failure time, Cox proportional hazards and probit models. We offer a theoretical treatment that includes local and non-local priors, and a general non-linear effect decomposition to improve power-sparsity trade-offs. We discuss a fundamental question: what solution can one hope to obtain when (inevitab…

Efficient Robust Proper Learning of Log-concave Distributions

We study the {\em robust proper learning} of univariate log-concave distributions (over continuous and discrete domains). Given a set of samples drawn from an unknown target distribution, we want to compute a log-concave hypothesis distribution that is as close as possible to the target, in total variation distance. In this work, we give the first computationally efficient algorithm for this learning problem. Our algorithm achieves the information-theoretically optimal sample size (up to a constant factor), runs in polynomial time, and is robust to model misspecification with nearly-optimal er…

Multivariate Log-Concave Distributions as a Nearly Parametric Model

In this paper we show that the family P_d of probability distributions on R^d with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. Hence the nonparametric model P_d has similar properties as parametric models such as, for instance, the family of all d-variate Gaussian distributions.

CAT(0)卦限空间中非参数密度估计量的一致性

Consistency of Nonparametric Density Estimators in CAT(0) Orthant Space

针对系统发育树分析用的CAT(0)空间中非参数密度估计量理论性质不明的问题，本文在更通用的CAT(0)卦限空间中证明了核密度估计与对数凹最大似然估计的一致性，校正了核估计的边界偏差。

基于 $\mathbb{L}_{1}$-损失的稳健密度估计。在直线上满足形状约束的密度估计的应用

Robust density estimation with the $\mathbb{L}_{1}$-loss. Applications to the estimation of a density on the line satisfying a shape constraint

本文提出一种基于 $\mathbb{L}_{1}$-损失和密度模型的稳健密度估计方法，适用于 MLE 不存在的模型，在全局收敛速度和自适应性质上具有与 MLE 类似的最优性，并对模型误设、污染和离群点具有稳健性。