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基于最优传输的多元分布无关检验的 Pitman 效率下界AI 解读

Abstract The Wilcoxon rank sum test is one of the most popular distribution-free two-sample tests for univariate data. Among the important reasons for their popularity are the striking results of Hodges–Lehmann and Chernoff–Savage, where the authors show that the asymptotic (Pitman) relative efficiency of Wilcoxon’s test compared to Student’s t-test, never falls below 0.864 (with identity score) and 1 (with Gaussian score), respectively. Motivated by these results, we…

随机试验中 Rosenbaum 秩估计量的稳健性与效率：设计视角AI 解读

Summary Mean-based estimators of causal effects in randomized experiments may behave poorly if the potential outcomes have a heavy tail or contain outliers. An alternative estimator proposed by Rosenbaum (1993) estimates a constant additive treatment effect by inverting a randomization test using ranks. We develop a design-based asymptotic theory for this rank-based estimator and study its robustness and efficiency properties. We show that Rosenbaum’s estimator is robust…

多元对称下基于最优传输的分布无关符号、秩与单样本位置检验AI 解读

We propose a novel and unified framework for distribution-free testing under multivariate symmetry (that includes central symmetry, sign symmetry, spherical symmetry, etc.) based on the theory of optimal transport. Our approach leads to notions of distribution-free generalized multivariate signs, absolute ranks and signed-ranks. As a consequence, we develop analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their…

通过非参数极大似然实现多元异方差经验贝叶斯估计AI 解读

Abstract Multivariate, heteroscedastic errors complicate statistical inference in many large-scale denoizing problems. Empirical Bayes is attractive in such settings, but standard parametric approaches rest on assumptions about the form of the prior distribution which can be hard to justify and which introduce unnecessary tuning parameters. We extend the nonparametric maximum-likelihood estimator (NPMLE) for Gaussian location mixture densities to allow for multivariate,…

经验部分贝叶斯多重检验与复合卡方决策AI 解读

A common task in high-throughput biology is to screen for associations across thousands of units of interest, for example, genes or proteins. Often, the data for each unit are modeled as Gaussian measurements with unknown mean and variance and are summarized as per-unit sample averages and sample variances. The downstream goal is multiple testing for the means. In this domain, it is routine to “moderate” (i.e., to shrink) the sample variances through parametric empirical…

多维凸回归：最小二乘估计量的次优性AI 解读

Under the usual nonparametric regression model with Gaussian errors, Least Squares Estimators (LSEs) over natural subclasses of convex functions are shown to be suboptimal for estimating a d-dimensional convex function in squared error loss when the dimension d is 5 or larger. The specific function classes considered include: (i) bounded convex functions supported on a polytope (in random design), (ii) Lipschitz convex functions supported on any convex domain (in random…

多个分布之间的核差异度量AI 解读

Given M≥2 distributions defined on a general measurable space, we introduce a nonparametric (kernel) measure of multi-sample dissimilarity (KMD)—a parameter that quantifies the difference between the M distributions. The population KMD, which takes values between 0 and 1, is 0 if and only if all the M distributions are the same, and 1 if and only if all the distributions are mutually singular. Moreover, KMD possesses many properties commonly associated with f-divergences…

凸性约束模型中局部参数的统计推断AI 解读

In this article, we develop automated inference methods for “local” parameters in a collection of convexity constrained models based on the natural constrained tuning-free estimators. A canonical example is given by the univariate convex regression model, in which automated inference is drawn for the function value, the function derivative at a fixed interior point, and the anti-mode of the convex regression function, based on the widely used tuning-free, piecewise linear…

凸性约束单指标模型中的半参数效率AI 解读

We consider estimation and inference in a single-index regression model with an unknown convex link function. We introduce a convex and Lipschitz constrained least-square estimator (CLSE) for both the parametric and the nonparametric components given independent and identically distributed observations. We prove the consistency and find the rates of convergence of the CLSE when the errors are assumed to have only q≥2 moments and are allowed to depend on the covariates. When…

基于测度输运的多元秩无分布非参数检验AI 解读

In this article, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of measure transportation. Unlike other existing proposals in the literature, these multivariate ranks share a number of useful properties with the usual one-dimensional ranks; most importantly, these ranks are distribution-free. This crucial observation allows us to design nonparametric tests that…

含协变量的双成分混合模型AI 解读

In this article, we study a generalization of the two-groups model in the presence of covariates—a problem that has recently received much attention in the statistical literature due to its applicability in multiple hypotheses testing problems. The model we consider allows for infinite dimensional parameters and offers flexibility in modeling the dependence of the response on the covariates. We discuss the identifiability issues arising in this model and systematically study…

最优传输定义的多元秩与分位数：一致性、速率和非参数检验AI 解读

In this paper, we study multivariate ranks and quantiles, defined using the theory of optimal transport, and build on the work of Chernozhukov et al. (Ann. Statist. 45 (2017) 223–256) and Hallin et al. (Ann. Statist. 49 (2021) 1139–1165). We study the characterization, computation and properties of the multivariate rank and quantile functions and their empirical counterparts. We derive the uniform consistency of these empirical estimates to their population versions, under…

凸约束高斯序列模型中似然比检验的高维渐近理论AI 解读

In the Gaussian sequence model Y=μ+ξ, we study the likelihood ratio test (LRT) for testing H0:μ=μ0 versus H1:μ∈K, where μ0∈K, and K is a closed convex set in Rn. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair (μ0,K), in the high-dimensional regime where the estimation error of the associated least squares estimator diverges in an appropriate sense. The normal approximation further…

通过完全单调性与Hardy–Krause变差推广多元保序回归和总变差去噪AI 解读

We consider the problem of nonparametric regression when the covariate is d dimensional, where d≥1. In this paper, we introduce and study two nonparametric least squares estimators (LSEs) in this setting—the entirely monotonic LSE and the constrained Hardy–Krause variation LSE. We show that these two LSEs are natural generalizations of univariate isotonic regression and univariate total variation denoising, respectively, to multiple dimensions. We discuss the…

一元全变差去噪与趋势滤波的自适应风险界AI 解读

We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given integer $r\geq1$, the $r$th order trend filtering estimator is defined as the minimizer of the sum of squared errors when we constrain (or penalize) the sum of the absolute $r$th order discrete derivatives of the fitted function at the design points. For $r=1$, the estimator reduces to total variation regularization which has received much attention in the statistics and…

投影估计量的自由度及其在多元非参数回归中的应用AI 解读

–In this article, we consider the nonparametric regression problem with multivariate predictors. We provide a characterization of the degrees of freedom and divergence for estimators of the unknown regression function, which are obtained as outputs of linearly constrained quadratic optimization procedures; namely, minimizers of the least-squares criterion with linear constraints and/or quadratic penalties. As special cases of our results, we derive explicit expressions for…

非标准问题中的分治方法与超效率现象AI 解读

We study how the divide and conquer principle works in non-standard problems where rates of convergence are typically slower than $\sqrt{n}$ and limit distributions are non-Gaussian, and provide a detailed treatment for a variety of important and well-studied problems involving nonparametric estimation of a monotone function. We find that for a fixed model, the pooled estimator, obtained by averaging nonstandard estimates across mutually exclusive subsamples, outperforms the…

多元凸回归及其变体的计算框架AI 解读

We study the nonparametric least squares estimator (LSE) of a multivariate convex regression function. The LSE, given as the solution to a quadratic program with O(n2) linear constraints (n being the sample size), is difficult to compute for large problems. Exploiting problem specific structure, we propose a scalable algorithmic framework based on the augmented Lagrangian method to compute the LSE. We develop a novel approach to obtain smooth convex approximations to the…

最大得分估计量的一致自助法AI 解读

In this paper we propose a new model-based smoothed bootstrap procedure for making inference on the maximum score estimator of Manski (1975, 1985) and prove its consistency. We provide a set of sufficient conditions for the consistency of any bootstrap procedure in this problem. We compare the finite sample performance of different bootstrap procedures through simulation studies. The results indicate that our proposed smoothed bootstrap outperforms other bootstrap schemes,…

利用形状约束估计思想检验线性回归模型AI 解读

Summary A formal likelihood ratio hypothesis test for the validity of a parametric regression function is proposed, using a large dimensional, non-parametric double-cone alternative. For example, the test against a constant function uses the alternative of increasing or decreasing regression functions, and the test against a linear function uses the convex or concave alternative. The test proposed is exact and unbiased and the critical value is easily computed. The power of…

双成分混合模型的估计及其多重检验应用AI 解读

Summary We consider a two-component mixture model with one known component. We develop methods for estimating the mixing proportion and the unknown distribution non-parametrically, given independent and identically distributed data from the mixture model, using ideas from shape-restricted function estimation. We establish the consistency of our estimators. We find the rate of convergence and asymptotic limit of the estimator for the mixing proportion. Completely automated…

保序及其他形状约束回归中的风险界AI 解读

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 $\mathbb{R}^{n}$. Under this setting, we prove bounds for the risk of the least squares estimator (LSE). The obtained risk bound behaves differently depending on the true sequence $\theta$ which highlights the adaptive behavior of $\theta$. As special cases of our general result, we derive risk…

回归中协变量的分位位置变换AI 解读

The need for comparing two regression functions arises frequently in statistical applications. Comparison of the usual regression functions is not very meaningful in situations where the distributions and the ranges of the covariates are different for the populations. For instance, in econometric studies, the prices of commodities and people's incomes observed at different time points may not be on comparable scales due to inflation and other economic factors. In this…

多元凸回归函数的非参数最小二乘估计AI 解读

This paper deals with the consistency of the nonparametric least squares estimator of a convex regression function when the predictor is multidimensional. We characterize and discuss the computation of such an estimator via the solution of certain quadratic and linear programs. Mild sufficient conditions for the consistency of this estimator and its subdifferentials in fixed and stochastic design regression settings are provided.