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利用高频数据估计混合分数稳定过程AI 解读

The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Lévy processes, and fractional Brownian motion. For this reason, it may be regarded as a basic building block for continuous time models. We study a stylized model consisting of a superposition of independent linear fractional stable motions and our focus is on parameter estimation of the model. Applying an estimating equations approach, we construct estimators for…

稀疏信号下高维协方差检验的精确最优性AI 解读

This paper considers one-sample testing of a high-dimensional covariance matrix by deriving the detection boundary as a function of the signal sparsity and signal strength under the sparse alternative hypotheses. It first shows that the optimal detection boundary for testing sparse means is the minimax detection lower boundary for testing the covariance matrix. A multilevel thresholding test is proposed and is shown to be able to attain the detection lower boundary over a…

未观测协变量对协变量自适应随机实验的影响AI 解读

Covariate-adaptive randomization (CAR) is commonly implemented in clinical trials to balance observed covariates. Recent studies have demonstrated the advantages of CAR procedures in balancing covariates and improving the subsequent statistical analysis. Covariate balance is crucial, but it is not a panacea for the valid statistical inferences. If the response to a treatment interacts with some unobserved covariates, the conclusion drawn from a CAR experiment may be…

一般高斯设计下的 Lasso 及其假设检验应用AI 解读

The Lasso is a method for high-dimensional regression, which is now commonly used when the number of covariates p is of the same order or larger than the number of observations n. Classical asymptotic normality theory does not apply to this model due to two fundamental reasons: (1) The regularized risk is nonsmooth; (2) The distance between the estimator θˆ and the true parameters vector θ∗ cannot be neglected. As a consequence, standard perturbative arguments that are the…

完全随机缺失的最优非参数检验及其与相容性的联系AI 解读

Given a set of incomplete observations, we study the nonparametric problem of testing whether data are Missing Completely At Random (MCAR). Our first contribution is to characterise precisely the set of alternatives that can be distinguished from the MCAR null hypothesis. This reveals interesting and novel links to the theory of Fréchet classes (in particular, compatible distributions) and linear programming, that allow us to propose MCAR tests that are consistent against…

依赖重尾数据的极端回归推断AI 解读

Nonparametric inference on tail conditional quantiles and their least squares analogs, expectiles, remains limited to i.i.d. data. We develop a fully operational inferential theory for extreme conditional quantiles and expectiles in the challenging framework of α-mixing, conditional heavy-tailed data whose tail index may vary with covariate values. This requires a dedicated treatment to deal with data sparsity in the far tail of the response, in addition to handling…

自适应且稳健的多任务学习AI 解读

We study the multitask learning problem that aims to simultaneously analyze multiple data sets collected from different sources and learn one model for each of them. We propose a family of adaptive methods that automatically utilize possible similarities among those tasks while carefully handling their differences. We derive sharp statistical guarantees for the methods and prove their robustness against outlier tasks. Numerical experiments on synthetic and real data sets…

时变极值依赖结构的统计推断AI 解读

We analyze the extreme value dependence of independent, not necessarily identically distributed multivariate regularly varying random vectors. More specifically, we propose estimators of the spectral measure locally at some time point and of the spectral measures integrated over time. The uniform asymptotic normality of these estimators is proved under suitable nonparametric smoothness and regularity assumptions. We then use the process convergence of the integrated spectral…

高维正则化回归估计量的普适性AI 解读

The Convex Gaussian Min–Max Theorem (CGMT) has emerged as a prominent theoretical tool for analyzing the precise stochastic behavior of various statistical estimators in the so-called high-dimensional proportional regime, where the sample size and the signal dimension are of the same order. However, a well-recognized limitation of the existing CGMT machinery rests in its stringent requirement on the exact Gaussianity of the design matrix, therefore rendering the obtained…

从高维含噪数据学习低维非线性结构：积分算子方法AI 解读

We propose a kernel-spectral embedding algorithm for learning low-dimensional nonlinear structures from noisy and high-dimensional observations, where the data sets are assumed to be sampled from a nonlinear manifold model and corrupted by high-dimensional noise. The algorithm employs an adaptive bandwidth selection procedure which does not rely on prior knowledge of the underlying manifold. The obtained low-dimensional embeddings can be further utilized for downstream…

通过算法稳定性进行选择后推断AI 解读

When the target of statistical inference is chosen in a data-driven manner, the guarantees provided by classical theories vanish. We propose a solution to the problem of inference after selection by building on the framework of algorithmic stability, in particular its branch with origins in the field of differential privacy. Stability is achieved via randomization of selection and it serves as a quantitative measure that is sufficient to obtain nontrivial post-selection…

离线强化学习中的投影状态—动作平衡权重AI 解读

Off-policy evaluation is considered a fundamental and challenging problem in reinforcement learning (RL). This paper focuses on value estimation of a target policy based on pre-collected data generated from a possibly different policy, under the framework of infinite-horizon Markov decision processes. Motivated by the recently developed marginal importance sampling method in RL and the covariate balancing idea in causal inference, we propose a novel estimator with…

凸约束下含噪线性逆问题的高维精确风险渐近AI 解读

In the standard Gaussian linear measurement model Y=Xμ0+ξ∈Rm with a fixed noise level σ>0, we consider the problem of estimating the unknown signal μ0 under a convex constraint μ0∈K, where K is a closed convex set in Rn. We show that the risk of the natural convex constrained least squares estimator (LSE) μˆ(σ) can be characterized exactly in high-dimensional limits, by that of the convex constrained LSE μˆKseq in the corresponding Gaussian sequence model at a different…

序贯可忽略性下部分可观测马尔可夫决策过程的离策略评估AI 解读

We consider off-policy evaluation of dynamic treatment rules under sequential ignorability, given an assumption that the underlying system can be modeled as a partially observed Markov decision process (POMDP). We propose an estimator, partial history importance weighting, and show that it can consistently estimate the stationary mean rewards of a target policy, given long enough draws from the behavior policy. We provide an upper bound on its error that decays polynomially…

信号去噪的交叉验证框架：趋势滤波、二分 CART 等应用AI 解读

This paper formulates a general cross-validation framework for signal denoising. The general framework is then applied to nonparametric regression methods such as trend filtering and dyadic CART. The resulting cross-validated versions are then shown to attain nearly the same rates of convergence as are known for the optimally tuned analogues. There did not exist any previous theoretical analyses of cross-validated versions of trend filtering or dyadic CART. To illustrate the…

偏差与方差权衡的下界AI 解读

It is a common phenomenon that for high-dimensional and nonparametric statistical models, rate-optimal estimators balance squared bias and variance. Although this balancing is widely observed, little is known whether methods exist that could avoid the trade-off between bias and variance. We propose a general strategy to obtain lower bounds on the variance of any estimator with bias smaller than a prespecified bound. This shows to which extent the bias-variance trade-off is…

持久贝蒂数及其他稳定化统计量的自助法AI 解读

We investigate multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. The work relates to other general results in the area of stabilizing statistics, including central limit theorems for geometric and topological functionals of Poisson and binomial processes in the critical regime, where limit theorems prove difficult to use in practice, motivating the use of a bootstrap approach. A smoothed bootstrap…

非平稳时间序列的图模型AI 解读

We propose NonStGM, a general nonparametric graphical modeling framework, for studying dynamic associations among the components of a nonstationary multivariate time series. It builds on the framework of Gaussian graphical models (GGM) and stationary time series graphical models (StGM) and complements existing works on parametric graphical models based on change point vector autoregressions (VAR). Analogous to StGM, the proposed framework captures conditional noncorrelations…

放宽独立同分布假设：通过平方根熵正则化实现自适应极小极大遗憾AI 解读

We consider prediction with expert advice when data are generated from distributions varying arbitrarily within an unknown constraint set. This semi-adversarial setting includes (at the extremes) the classical i.i.d. setting, when the unknown constraint set is restricted to be a singleton, and the unconstrained adversarial setting, when the constraint set is the set of all distributions. The Hedge algorithm—long known to be minimax (rate) optimal in the adversarial…

单指标 Fréchet 回归AI 解读

Single index models provide an effective dimension reduction tool in regression, especially for high-dimensional data, by projecting a general multivariate predictor onto a direction vector. We propose a novel single-index model for regression models where metric space-valued random object responses are coupled with multivariate Euclidean predictors. The responses in this regression model include complex, non-Euclidean data, including covariance matrices, graph Laplacians of…

连接因子模型与稀疏模型AI 解读

Factor and sparse models are widely used to impose a low-dimensional structure in high-dimensions. However, they are seemingly mutually exclusive. We propose a lifting method that combines the merits of these two models in a supervised learning methodology that allows for efficiently exploring all the information in high-dimensional datasets. The method is based on a flexible model for high-dimensional panel data with observable and/or latent common factors and idiosyncratic…

最优变点检测与定位AI 解读

Given a times series Y in Rn, with a piecewise constant mean and independent components, the twin problems of change-point detection and change-point localization, respectively amount to detecting the existence of times where the mean varies and estimating the positions of those change-points. In this work, we tightly characterize optimal rates for both problems and uncover the phase transition phenomenon from a global testing problem to a local estimation problem.…

具有对抗性响应的普遍回归AI 解读

We provide algorithms for regression with adversarial responses under large classes of non-i.i.d. instance sequences, on general separable metric spaces, with provably minimal assumptions. We also give characterizations of learnability in this regression context. We consider universal consistency, which asks for strong consistency of a learner without restrictions on the value responses. Our analysis shows that such an objective is achievable for a significantly larger class…

贝叶斯多元保序回归中可信区间的覆盖率AI 解读

We consider the nonparametric multivariate isotonic regression problem, where the regression function is assumed to be nondecreasing with respect to each predictor. Our goal is to construct a Bayesian credible interval for the function value at a given interior point with assured limiting frequentist coverage. A natural prior on the regression function is given by a random step function with a suitable prior on increasing step-heights, but the resulting posterior…