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非参数贝叶斯程序中的 Bernstein–von Mises 现象AI 解读

We continue the investigation of Bernstein–von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999–2028]. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined, and prove Bernstein–von Mises theorems for a variety of priors in the setting of Gaussian nonparametric regression and in the i.i.d. sampling model. From these results we deduce several applications where posterior-based inference coincides…

半参数高斯 Copula 模型：几何与有效秩估计AI 解读

We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize efficiency of the…

L²损失下密度估计的最优交叉验证AI 解读

We analyze the performance of cross-validation (CV) in the density estimation framework with two purposes: (i) risk estimation and (ii) model selection. The main focus is given to the so-called leave-$p$-out CV procedure (Lpo), where $p$ denotes the cardinality of the test set. Closed-form expressions are settled for the Lpo estimator of the risk of projection estimators. These expressions provide a great improvement upon $V$-fold cross-validation in terms of variability and…

Bernstein–von Mises 定理与非正则模型AI 解读

We study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the “true” solution occurs on the boundary of the parameter space. We show that in this case Bayesian inference is consistent, and that the posterior distribution has not only Gaussian components as in the case of regular models (the Bernstein–von Mises theorem) but also has Gamma distribution components whose form depends on the behaviour of the prior distribution…

超高维纵向数据的非参数独立筛选与结构识别AI 解读

Ultra-high dimensional longitudinal data are increasingly common and the analysis is challenging both theoretically and methodologically. We offer a new automatic procedure for finding a sparse semivarying coefficient model, which is widely accepted for longitudinal data analysis. Our proposed method first reduces the number of covariates to a moderate order by employing a screening procedure, and then identifies both the varying and constant coefficients using a group SCAD…

反集中性与诚实、自适应置信带AI 解读

Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the classical Smirnov–Bickel–Rosenblatt (SBR) condition; see, for example, Giné and Nickl [Probab. Theory Related Fields 143 (2009) 569–596]. This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance…

通过切片逆回归进行一般指标模型的变量选择AI 解读

Variable selection, also known as feature selection in machine learning, plays an important role in modeling high dimensional data and is key to data-driven scientific discoveries. We consider here the problem of detecting influential variables under the general index model, in which the response is dependent of predictors through an unknown function of one or more linear combinations of them. Instead of building a predictive model of the response given combinations of…

一般正交阵列空间填充设计的中心极限定理AI 解读

Orthogonal array based space-filling designs (Owen [Statist. Sinica 2 (1992a) 439–452]; Tang [J. Amer. Statist. Assoc. 88 (1993) 1392–1397]) have become popular in computer experiments, numerical integration, stochastic optimization and uncertainty quantification. As improvements of ordinary Latin hypercube designs, these designs achieve stratification in multi-dimensions. If the underlying orthogonal array has strength $t$, such designs achieve uniformity up to $t$ dimension…

局部病例—对照抽样：类别不平衡数据的高效子抽样AI 解读

For classification problems with significant class imbalance, subsampling can reduce computational costs at the price of inflated variance in estimating model parameters. We propose a method for subsampling efficiently for logistic regression by adjusting the class balance locally in feature space via an accept–reject scheme. Our method generalizes standard case-control sampling, using a pilot estimate to preferentially select examples whose responses are conditionally rare…

大规模 t 检验中错误发现率控制的相变与正则化自助法AI 解读

Applying the Benjamini and Hochberg (B–H) method to multiple Student’s $t$ tests is a popular technique for gene selection in microarray data analysis. Given the nonnormality of the population, the true $p$-values of the hypothesis tests are typically unknown. Hence it is common to use the standard normal distribution $N(0,1)$, Student’s $t$ distribution $t_{n-1}$ or the bootstrap method to estimate the $p$-values. In this paper, we prove that when the population has the…

用于充分降维的自适应复合分位数方法AI 解读

Sufficient dimension reduction [J. Amer. Statist. Assoc. 86 (1991) 316–342] has long been a prominent issue in multivariate nonparametric regression analysis. To uncover the central dimension reduction space, we propose in this paper an adaptive composite quantile approach. Compared to existing methods, (1) it requires minimal assumptions and is capable of revealing all dimension reduction directions; (2) it is robust against outliers and (3) it is structure-adaptive, thus…

一致弱收敛失效时：借助上图与下图研究依赖函数和残差经验过程AI 解读

In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in…

经验过程上确界的高斯近似AI 解读

This paper develops a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the sup-norm. We prove an abstract approximation theorem applicable to a wide variety of statistical problems, such as construction of uniform confidence bands for functions. Notably, the bound in the main approximation theorem is nonasymptotic and the theorem…

非参数密度脊估计AI 解读

We study the problem of estimating the ridges of a density function. Ridge estimation is an extension of mode finding and is useful for understanding the structure of a density. It can also be used to find hidden structure in point cloud data. We show that, under mild regularity conditions, the ridges of the kernel density estimator consistently estimate the ridges of the true density. When the data are noisy measurements of a manifold, we show that the ridges are close and…

In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different $\pi$-reversible Markov transition kernels $P$ and $Q$. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels $P_{i}$ and $Q_{i}$, $i\in\{0,1\}$, as soon as the kernels of each pair $(P_{0},P_{1})$ and $(Q_{0},Q_{1})$…

离散数据的马尔可夫无环有向混合图AI 解读

Acyclic directed mixed graphs (ADMGs) are graphs that contain directed ($\rightarrow$) and bidirected ($\leftrightarrow$) edges, subject to the constraint that there are no cycles of directed edges. Such graphs may be used to represent the conditional independence structure induced by a DAG model containing hidden variables on its observed margin. The Markovian model associated with an ADMG is simply the set of distributions obeying the global Markov property, given via a…

含噪观测下二次协变差矩阵的估计：局部矩方法与效率AI 解读

An efficient estimator is constructed for the quadratic covariation or integrated co-volatility matrix of a multivariate continuous martingale based on noisy and nonsynchronous observations under high-frequency asymptotics. Our approach relies on an asymptotically equivalent continuous-time observation model where a local generalised method of moments in the spectral domain turns out to be optimal. Asymptotic semi-parametric efficiency is established in the Cramér–Rao sense.…

单指标调制的多重检验AI 解读

In the context of large-scale multiple testing, hypotheses are often accompanied with certain prior information. In this paper, we present a single-index modulated (SIM) multiple testing procedure, which maintains control of the false discovery rate while incorporating prior information, by assuming the availability of a bivariate $p$-value, $(p_{1},p_{2})$, for each hypothesis, where $p_{1}$ is a preliminary $p$-value from prior information and $p_{2}$ is the primary $p$-val…

二阶有效的经验贝叶斯置信区间AI 解读

We introduce a new adjusted residual maximum likelihood method (REML) in the context of producing an empirical Bayes (EB) confidence interval for a normal mean, a problem of great interest in different small area applications. Like other rival empirical Bayes confidence intervals such as the well-known parametric bootstrap empirical Bayes method, the proposed interval is second-order correct, that is, the proposed interval has a coverage error of order $O(m^{-{3}/{2}})$.…

用马尔可夫跳跃过程建模带重组的溯祖过程AI 解读

Genetic recombination is one of the most important mechanisms that can generate and maintain diversity, and recombination information plays an important role in population genetic studies. However, the phenomenon of recombination is extremely complex, and hence simulation methods are indispensable in the statistical inference of recombination. So far there are mainly two classes of simulation models practically in wide use: back-in-time models and spatially moving models.…

存在跳跃时积分波动率估计收敛速度的一点说明AI 解读

The optimal rate of convergence of estimators of the integrated volatility, for a discontinuous Itô semimartingale sampled at regularly spaced times and over a fixed time interval, has been a long-standing problem, at least when the jumps are not summable. In this paper, we study this optimal rate, in the minimax sense and for appropriate “bounded” nonparametric classes of semimartingales. We show that, if the $r$th powers of the jumps are summable for some $r\in[0,2)$, the…

无限维空间中的空间分布、分位数与深度AI 解读

The spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data. In this paper, we investigate the concept of spatial distribution for data in infinite dimensional Banach spaces. Many technical difficulties are encountered in such spaces that are primarily due to the noncompactness of the closed unit ball. In this work, we prove some Glivenko–Cantelli and Donsker-type results for the empirical spatial…

高维模型中渐近最优的置信区域与检验AI 解读

We propose a general method for constructing confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in a high-dimensional model. It can be easily adjusted for multiplicity taking dependence among tests into account. For linear models, our method is essentially the same as in Zhang and Zhang [J. R. Stat. Soc. Ser. B Stat. Methodol. 76 (2014) 217–242]: we analyze its asymptotic properties and establish its asymptotic…

非参数扩散与欧拉离散格式统计实验的渐近等价性AI 解读

We prove a global asymptotic equivalence of experiments in the sense of Le Cam’s theory. The experiments are a continuously observed diffusion with nonparametric drift and its Euler scheme. We focus on diffusions with nonconstant-known diffusion coefficient. The asymptotic equivalence is proved by constructing explicit equivalence mappings based on random time changes. The equivalence of the discretized observation of the diffusion and the corresponding Euler scheme…