在本文中，我们提出了一种多个内核测试程序，以推断几个因素（例如不同的治疗组，性别，病史）及其相互作用同时引起了人们的兴趣。我们的方法能够处理复杂的数据，并且当假设诸如相称性不能合理时，可以看作是无所不在的COX模型的替代方法。我们的方法结合了来自生存分析，机器学习和多次测试的众所周知的概念：加权的对数秩检验，内核方法和多个对比度测试。这样，可以检测到超出经典比例危害设置以外的复杂危险替代方案。此外，通过充分利用单个测试程序的依赖性结构以避免功率损失来进行多个比较。总的来说，这为阶乘生存设计提供了灵活而强大的程序，其理论有效性通过Martingale论证和$ v $统计的理论证明。我们在广泛的仿真研究中评估了方法的性能，并通过真实的数据分析对其进行了说明。

基于内核的测试提供了一个简单而有效的框架，该框架使用繁殖内核希尔伯特空间的理论设计非参数测试程序。在本文中，我们提出了新的理论工具，可用于在几种数据方案以及许多不同的测试问题中研究基于内核测试的渐近行为。与当前的方法不同，我们的方法避免使用冗长的$ u $和$ v $统计信息扩展并限制定理，该定理通常出现在文献中，并直接与希尔伯特空格上的随机功能合作。因此，我们的框架会导致对内核测试的简单明了的分析，只需要轻度的规律条件。此外，我们表明，通常可以通过证明我们方法所需的规律条件既足够又需要进行必要的规律条件来改进我们的分析。为了说明我们的方法的有效性，我们为有条件的独立性测试问题提供了一项新的内核测试，以及针对已知的基于内核测试的新分析。

我们在右审查的生存时间和协变量之间介绍一般的非参数独立测试，这可能是多变量的。我们的测试统计数据具有双重解释，首先是潜在无限的重量索引日志秩检验的超级索引，具有属于函数的再现内核HILBERT空间（RKHS）的重量函数;其次，作为某些有限措施的嵌入差异的规范，与Hilbert-Schmidt独立性标准（HSIC）测试统计类似。我们研究了测试的渐近性质，找到了足够的条件，以确保我们的测试在任何替代方案下正确拒绝零假设。可以直截了当地计算测试统计，并且通过渐近总体的野外自注程序进行拒绝阈值。对模拟和实际数据的广泛调查表明，我们的测试程序通常比检测复杂的非线性依赖的竞争方法更好。

Classical asymptotic theory for statistical inference usually involves calibrating a statistic by fixing the dimension $d$ while letting the sample size $n$ increase to infinity. Recently, much effort has been dedicated towards understanding how these methods behave in high-dimensional settings, where $d$ and $n$ both increase to infinity together. This often leads to different inference procedures, depending on the assumptions about the dimensionality, leaving the practitioner in a bind: given a dataset with 100 samples in 20 dimensions, should they calibrate by assuming $n \gg d$, or $d/n \approx 0.2$? This paper considers the goal of dimension-agnostic inference; developing methods whose validity does not depend on any assumption on $d$ versus $n$. We introduce an approach that uses variational representations of existing test statistics along with sample splitting and self-normalization to produce a new test statistic with a Gaussian limiting distribution, regardless of how $d$ scales with $n$. The resulting statistic can be viewed as a careful modification of degenerate U-statistics, dropping diagonal blocks and retaining off-diagonal blocks. We exemplify our technique for some classical problems including one-sample mean and covariance testing, and show that our tests have minimax rate-optimal power against appropriate local alternatives. In most settings, our cross U-statistic matches the high-dimensional power of the corresponding (degenerate) U-statistic up to a $\sqrt{2}$ factor.

Testing the significance of a variable or group of variables $X$ for predicting a response $Y$, given additional covariates $Z$, is a ubiquitous task in statistics. A simple but common approach is to specify a linear model, and then test whether the regression coefficient for $X$ is non-zero. However, when the model is misspecified, the test may have poor power, for example when $X$ is involved in complex interactions, or lead to many false rejections. In this work we study the problem of testing the model-free null of conditional mean independence, i.e. that the conditional mean of $Y$ given $X$ and $Z$ does not depend on $X$. We propose a simple and general framework that can leverage flexible nonparametric or machine learning methods, such as additive models or random forests, to yield both robust error control and high power. The procedure involves using these methods to perform regressions, first to estimate a form of projection of $Y$ on $X$ and $Z$ using one half of the data, and then to estimate the expected conditional covariance between this projection and $Y$ on the remaining half of the data. While the approach is general, we show that a version of our procedure using spline regression achieves what we show is the minimax optimal rate in this nonparametric testing problem. Numerical experiments demonstrate the effectiveness of our approach both in terms of maintaining Type I error control, and power, compared to several existing approaches.

In nonparametric independence testing, we observe i.i.d.\ data $\{(X_i,Y_i)\}_{i=1}^n$, where $X \in \mathcal{X}, Y \in \mathcal{Y}$ lie in any general spaces, and we wish to test the null that $X$ is independent of $Y$. Modern test statistics such as the kernel Hilbert-Schmidt Independence Criterion (HSIC) and Distance Covariance (dCov) have intractable null distributions due to the degeneracy of the underlying U-statistics. Thus, in practice, one often resorts to using permutation testing, which provides a nonasymptotic guarantee at the expense of recalculating the quadratic-time statistics (say) a few hundred times. This paper provides a simple but nontrivial modification of HSIC and dCov (called xHSIC and xdCov, pronounced ``cross'' HSIC/dCov) so that they have a limiting Gaussian distribution under the null, and thus do not require permutations. This requires building on the newly developed theory of cross U-statistics by Kim and Ramdas (2020), and in particular developing several nontrivial extensions of the theory in Shekhar et al. (2022), which developed an analogous permutation-free kernel two-sample test. We show that our new tests, like the originals, are consistent against fixed alternatives, and minimax rate optimal against smooth local alternatives. Numerical simulations demonstrate that compared to the full dCov or HSIC, our variants have the same power up to a $\sqrt 2$ factor, giving practitioners a new option for large problems or data-analysis pipelines where computation, not sample size, could be the bottleneck.

The kernel Maximum Mean Discrepancy~(MMD) is a popular multivariate distance metric between distributions that has found utility in two-sample testing. The usual kernel-MMD test statistic is a degenerate U-statistic under the null, and thus it has an intractable limiting distribution. Hence, to design a level-$\alpha$ test, one usually selects the rejection threshold as the $(1-\alpha)$-quantile of the permutation distribution. The resulting nonparametric test has finite-sample validity but suffers from large computational cost, since every permutation takes quadratic time. We propose the cross-MMD, a new quadratic-time MMD test statistic based on sample-splitting and studentization. We prove that under mild assumptions, the cross-MMD has a limiting standard Gaussian distribution under the null. Importantly, we also show that the resulting test is consistent against any fixed alternative, and when using the Gaussian kernel, it has minimax rate-optimal power against local alternatives. For large sample sizes, our new cross-MMD provides a significant speedup over the MMD, for only a slight loss in power.

尽管U统计量在现代概率和统计学中存在着无处不在的，但其在依赖框架中的非反应分析可能被忽略了。在最近的一项工作中，已经证明了对统一的马尔可夫链的U级统计数据的新浓度不平等。在本文中，我们通过在三个不同的研究领域中进一步推动了当前知识状态，将这一理论突破付诸实践。首先，我们为使用MCMC方法估算痕量类积分运算符光谱的新指数不平等。新颖的是，这种结果适用于具有正征和负征值的内核，据我们所知，这是新的。此外，我们研究了使用成对损失函数和马尔可夫链样品的在线算法的概括性能。我们通过展示如何从任何在线学习者产生的假设序列中提取低风险假设来提供在线到批量转换结果。我们最终对马尔可夫链的不变度度量的密度进行了拟合优度测试的非反应分析。我们确定了一些类别的替代方案，基于$ L_2 $距离的测试具有规定的功率。

We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD). We present two distributionfree tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

In high dimensional variable selection problems, statisticians often seek to design multiple testing procedures controlling the false discovery rate (FDR) and simultaneously discovering more relevant variables. Model-X methods, such as Knockoffs and conditional randomization tests, achieve the first goal of finite-sample FDR control under the assumption of known covariates distribution. However, it is not clear whether these methods can concurrently achieve the second goal of maximizing the number of discoveries. In fact, designing procedures to discover more relevant variables with finite-sample FDR control is a largely open question, even in the arguably simplest linear models. In this paper, we derive near-optimal testing procedures in high dimensional Bayesian linear models with isotropic covariates. We propose a Model-X multiple testing procedure, PoEdCe, which provably controls the frequentist FDR from finite samples even under model misspecification, and conjecturally achieves near-optimal power when the data follow the Bayesian linear model with a known prior. PoEdCe has three important ingredients: Posterior Expectation, distilled Conditional randomization test (dCRT), and the Benjamini-Hochberg procedure with e-values (eBH). The optimality conjecture of PoEdCe is based on a heuristic calculation of its asymptotic true positive proportion (TPP) and false discovery proportion (FDP), which is supported by methods from statistical physics as well as extensive numerical simulations. Furthermore, when the prior is unknown, we show that an empirical Bayes variant of PoEdCe still has finite-sample FDR control and achieves near-optimal power.

我们提出了一种基于最大平均差异（MMD）的新型非参数两样本测试，该测试是通过具有不同核带宽的聚合测试来构建的。这种称为MMDAGG的聚合过程可确保对所使用的内核的收集最大化测试能力，而无需持有核心选择的数据（这会导致测试能力损失）或任意内核选择，例如中位数启发式。我们在非反应框架中工作，并证明我们的聚集测试对Sobolev球具有最小自适应性。我们的保证不仅限于特定的内核，而是符合绝对可集成的一维翻译不变特性内核的任何产品。此外，我们的结果适用于流行的数值程序来确定测试阈值，即排列和野生引导程序。通过对合成数据集和现实世界数据集的数值实验，我们证明了MMDAGG优于MMD内核适应的替代方法，用于两样本测试。

We study non-parametric estimation of the value function of an infinite-horizon $\gamma$-discounted Markov reward process (MRP) using observations from a single trajectory. We provide non-asymptotic guarantees for a general family of kernel-based multi-step temporal difference (TD) estimates, including canonical $K$-step look-ahead TD for $K = 1, 2, \ldots$ and the TD$(\lambda)$ family for $\lambda \in [0,1)$ as special cases. Our bounds capture its dependence on Bellman fluctuations, mixing time of the Markov chain, any mis-specification in the model, as well as the choice of weight function defining the estimator itself, and reveal some delicate interactions between mixing time and model mis-specification. For a given TD method applied to a well-specified model, its statistical error under trajectory data is similar to that of i.i.d. sample transition pairs, whereas under mis-specification, temporal dependence in data inflates the statistical error. However, any such deterioration can be mitigated by increased look-ahead. We complement our upper bounds by proving minimax lower bounds that establish optimality of TD-based methods with appropriately chosen look-ahead and weighting, and reveal some fundamental differences between value function estimation and ordinary non-parametric regression.

We provide results that exactly quantify how data augmentation affects the convergence rate and variance of estimates. They lead to some unexpected findings: Contrary to common intuition, data augmentation may increase rather than decrease the uncertainty of estimates, such as the empirical prediction risk. Our main theoretical tool is a limit theorem for functions of randomly transformed, high-dimensional random vectors. The proof draws on work in probability on noise stability of functions of many variables. The pathological behavior we identify is not a consequence of complex models, but can occur even in the simplest settings -- one of our examples is a ridge regressor with two parameters. On the other hand, our results also show that data augmentation can have real, quantifiable benefits.

本文衍生了置信区间（CI）和时间统一的置信序列（CS），用于从有限观测值中估算未知平均值的经典问题。我们提出了一种衍生浓度界限的一般方法，可以看作是著名的切尔诺夫方法的概括（和改进）。它的核心是基于推导一类新的复合非负胸腔，通过投注和混合方法与测试的连接很强。我们展示了如何将这些想法扩展到无需更换的情况下，这是另一个经过深入研究的问题。在所有情况下，我们的界限都适应未知的差异，并且基于Hoeffding或经验的Bernstein不平等及其最近的Supermartingale概括，经验上大大优于现有方法。简而言之，我们为四个基本问题建立了一个新的最先进的问题：在有或没有替换的情况下进行采样时，CS和CI进行有限的手段。

统计推断中的主要范式取决于I.I.D.的结构。来自假设的无限人群的数据。尽管它取得了成功，但在复杂的数据结构下，即使在清楚无限人口所代表的内容的情况下，该框架在复杂的数据结构下仍然不灵活。在本文中，我们探讨了一个替代框架，在该框架中，推断只是对模型误差的不变性假设，例如交换性或符号对称性。作为解决这个不变推理问题的一般方法，我们提出了一个基于随机的过程。我们证明了该过程的渐近有效性的一般条件，并在许多数据结构中说明了，包括单向和双向布局中的群集误差。我们发现，通过残差随机化的不变推断具有三个吸引人的属性：（1）在弱且可解释的条件下是有效的，可以解决重型数据，有限聚类甚至一些高维设置的问题。 （2）它在有限样品中是可靠的，因为它不依赖经典渐近学所需的规律性条件。 （3）它以适应数据结构的统一方式解决了推断问题。另一方面，诸如OLS或Bootstrap之类的经典程序以I.I.D.为前提。结构，只要实际问题结构不同，就需要修改。经典框架中的这种不匹配导致了多种可靠的误差技术和自举变体，这些变体经常混淆应用研究。我们通过广泛的经验评估证实了这些发现。残留随机化对许多替代方案的表现有利，包括可靠的误差方法，自举变体和分层模型。

本文研究了基于Laplacian Eigenmaps（Le）的基于Laplacian EIGENMAPS（PCR-LE）的主要成分回归的统计性质，这是基于Laplacian Eigenmaps（Le）的非参数回归的方法。 PCR-LE通过投影观察到的响应的向量$ {\ bf y} =（y_1，\ ldots，y_n）$ to to changbood图表拉普拉斯的某些特征向量跨越的子空间。我们表明PCR-Le通过SoboLev空格实现了随机设计回归的最小收敛速率。在设计密度$ P $的足够平滑条件下，PCR-le达到估计的最佳速率（其中已知平方$ l ^ 2 $ norm的最佳速率为$ n ^ { - 2s /（2s + d） ）} $）和健美的测试（$ n ^ { - 4s /（4s + d）$）。我们还表明PCR-LE是\ EMPH {歧管Adaptive}：即，我们考虑在小型内在维度$ M $的歧管上支持设计的情况，并为PCR-LE提供更快的界限Minimax估计（$ n ^ { - 2s /（2s + m）$）和测试（$ n ^ { - 4s /（4s + m）$）收敛率。有趣的是，这些利率几乎总是比图形拉普拉斯特征向量的已知收敛率更快;换句话说，对于这个问题的回归估计的特征似乎更容易，统计上讲，而不是估计特征本身。我们通过经验证据支持这些理论结果。

我们在分布式框架中得出最小值测试错误，其中数据被分成多个机器，并且它们与中央机器的通信仅限于$ b $位。我们研究了高斯白噪声下的$ d $ - 和无限维信号检测问题。我们还得出达到理论下限的分布式测试算法。我们的结果表明，分布式测试受到从根本上不同的现象，这些现象在分布式估计中未观察到。在我们的发现中，我们表明，可以访问共享随机性的测试协议在某些制度中的性能比不进行的测试协议可以更好地表现。我们还观察到，即使仅使用单个本地计算机上可用的信息，一致的非参数分布式测试始终是可能的，即使只有$ 1 $的通信和相应的测试优于最佳本地测试。此外，我们还得出了自适应非参数分布测试策略和相应的理论下限。

由于其出色的经验表现，随机森林是过去十年中使用的机器学习方法之一。然而，由于其黑框的性质，在许多大数据应用中很难解释随机森林的结果。量化各个特征在随机森林中的实用性可以大大增强其解释性。现有的研究表明，一些普遍使用的特征对随机森林的重要性措施遭受了偏见问题。此外，对于大多数现有方法，缺乏全面的规模和功率分析。在本文中，我们通过假设检验解决了问题，并提出了一个自由化特征 - 弥散性相关测试（事实）的框架，以评估具有偏见性属性的随机森林模型中给定特征的重要性，我们零假设涉及该特征是否与所有其他特征有条件地独立于响应。关于高维随机森林一致性的一些最新发展，对随机森林推断的这种努力得到了赋予的能力。在存在功能依赖性的情况下，我们的事实测试的香草版可能会遇到偏见问题。我们利用偏置校正的不平衡和调节技术。我们通过增强功率的功能转换将合奏的想法进一步纳入事实统计范围。在相当普遍的具有依赖特征的高维非参数模型设置下，我们正式确定事实可以提供理论上合理的随机森林具有P值，并通过非催化分析享受吸引人的力量。新建议的方法的理论结果和有限样本优势通过几个模拟示例和与Covid-19的经济预测应用进行了说明。

套索是一种高维回归的方法，当时，当协变量$ p $的订单数量或大于观测值$ n $时，通常使用它。由于两个基本原因，经典的渐近态性理论不适用于该模型：$（1）$正规风险是非平滑的； $（2）$估算器$ \ wideHat {\ boldsymbol {\ theta}} $与true参数vector $ \ boldsymbol {\ theta}^*$无法忽略。结果，标准的扰动论点是渐近正态性的传统基础。另一方面，套索估计器可以精确地以$ n $和$ p $大，$ n/p $的订单为一。这种表征首先是在使用I.I.D的高斯设计的情况下获得的。协变量：在这里，我们将其推广到具有非偏差协方差结构的高斯相关设计。这是根据更简单的``固定设计''模型表示的。我们在两个模型中各种数量的分布之间的距离上建立了非反应界限，它们在合适的稀疏类别中均匀地固定在信号上$ \ boldsymbol {\ theta}^*$。作为应用程序，我们研究了借助拉索的分布，并表明需要校正程度对于计算有效的置信区间是必要的。

有许多可用于选择优先考虑治疗的可用方法，包括基于治疗效果估计，风险评分和手工制作规则的遵循申请。我们将秩加权平均治疗效应（RATY）指标作为一种简单常见的指标系列，用于比较水平竞争范围的治疗优先级规则。对于如何获得优先级规则，率是不可知的，并且仅根据他们在识别受益于治疗中受益的单位的方式进行评估。我们定义了一系列速率估算器，并证明了一个中央限位定理，可以在各种随机和观测研究环境中实现渐近精确的推断。我们为使用自主置信区间的使用提供了理由，以及用于测试关于治疗效果中的异质性的假设的框架，与优先级规则相关。我们对速率的定义嵌套了许多现有度量，包括QINI系数，以及我们的分析直接产生了这些指标的推论方法。我们展示了我们从个性化医学和营销的示例中的方法。在医疗环境中，使用来自Sprint和Accor-BP随机对照试验的数据，我们发现没有明显的证据证明异质治疗效果。另一方面，在大量的营销审判中，我们在一些数字广告活动的治疗效果中发现了具有的强大证据，并证明了如何使用率如何比较优先考虑估计风险的目标规则与估计治疗效益优先考虑的目标规则。