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多元独立性 Cramér–von Mises 检验的渐近局部效率AI 解读

Deheuvels [J. Multivariate Anal. 11 (1981) 102–113] and Genest and Rémillard [Test 13 (2004) 335–369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cramér–von Mises statistics derived from a Möbius decomposition of the empirical copula process. A result on the large-sample behavior of this process under contiguous sequences of alternatives is used here to give a representation of the limiting…

回归 MM 估计与约束 M 估计的最大偏差函数AI 解读

We derive the maximum bias functions of the MM-estimates and the constrained M-estimates or CM-estimates of regression and compare them to the maximum bias functions of the S-estimates and the τ-estimates of regression. In these comparisons, the CM-estimates tend to exhibit the most favorable bias-robustness properties. Also, under the Gaussian model, it is shown how one can construct a CM-estimate which has a smaller maximum bias function than a given S-estimate, that is,…

变系数分位数回归AI 解读

Quantile regression provides a framework for modeling statistical quantities of interest other than the conditional mean. The regression methodology is well developed for linear models, but less so for nonparametric models. We consider conditional quantiles with varying coefficients and propose a methodology for their estimation and assessment using polynomial splines. The proposed estimators are easy to compute via standard quantile regression algorithms and a stepwise knot…

函数型线性回归的方法与收敛速率AI 解读

In functional linear regression, the slope “parameter” is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of contact with a range of methodologies, including statistical smoothing and deconvolution. The standard approach to estimating the slope function is based explicitly on functional principal components analysis and, consequently, on spectral…

作者回复：局部 Rademacher 复杂度与风险最小化中的 oracle 不等式AI 解读

I would like to thank the discussants for a number of deep and interesting comments and for their inspiring work on the subject over the years.I will not be able to address all the issues raised in the discussion; I will concentrate just on several of them. Local complexities and excess risk bounds.The first question is about possible ways to define distribution-and data-dependent complexities (such as local Rademacher complexities).The approach taken in my paper is based on…

讨论：风险最小化中的局部 Rademacher 复杂度与 Oracle 不等式AI 解读

This paper unifies and extends important theoretical results on empirical risk minimization and model selection.It makes extensive and efficient use of new probability inequalities for the amount of concentration of the (possibly symmetrized) empirical process around its mean.The results are very subtle and very pleasing indeed, as they show that oracle inequalities exist for very general problems.There are in my view two aspects which need special attention.First, the paper…

讨论：局部 Rademacher 复杂度与风险最小化的预言不等式AI 解读

The paper of Vladimir Koltchinskii has been circulating around for several years and already has become an important reference in statistical learning theory.One of the main achievements of the paper (further abbreviated as [VK]) is to propose very general techniques of proving oracle inequalities for excess risk under a control of the variance, that is, for example, under conditions (6.1) or (6.2) (often called margin or low noise conditions) or similar assumptions in terms…

讨论：风险最小化中的局部Rademacher复杂度与预言机不等式AI 解读

Koltchinskii is to be congratulated for developing a unified framework.This elegant framework is general and allows a user to apply it directly instead of deriving bounds in each risk minimization problem.In the past decade, the problem of risk minimization has been extensively studied in function estimation and classification.In function estimation, it has been investigated using the empirical process technique under the name of minimum contrast or sieve estimation in, for…

讨论：风险最小化中的局部 Rademacher 复杂度与预言不等式AI 解读

In this magnificent paper, Professor Koltchinskii offers general and powerful performance bounds for empirical risk minimization, a fundamental principle of statistical learning theory.Since the elegant pioneering work of Vapnik and Chervonenkis in the early 1970s, various such bounds have been known that relate the performance of empirical risk minimizers to combinatorial and geometrical features of the class over which the minimization is performed.This area of research…

风险最小化中的局部拉德马赫复杂度与预言不等式AI 解读

Let ℱ be a class of measurable functions f:S↦[0, 1] defined on a probability space (S, $\mathcal{A}$, P). Given a sample (X1, …, Xn) of i.i.d. random variables taking values in S with common distribution P, let Pn denote the empirical measure based on (X1, …, Xn). We study an empirical risk minimization problem Pnf→min , f∈ℱ. Given a solution f̂n of this problem, the goal is to obtain very general upper bounds on its excess risk $$\mathcal{E}_{P}(\hat{f}_{n}):=P\hat{f}_{n}-\inf_{f\in \mathcal{F}}Pf,$$…

高效独立成分分析AI 解读

Independent component analysis (ICA) has been widely used for blind source separation in many fields, such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on M-estimates have been proposed for estimating the mixing matrix. Recently, several nonparametric methods have been developed, but in-depth analysis of asymptotic efficiency has not been available. We analyze ICA using semiparametric theories and propose a…

稀疏指数分布数据的错误发现率阈值估计的渐近极小极大性AI 解读

We apply FDR thresholding to a non-Gaussian vector whose coordinates Xi, i=1, …, n, are independent exponential with individual means μi. The vector μ=(μi) is thought to be sparse, with most coordinates 1 but a small fraction significantly larger than 1; roughly, most coordinates are simply ‘noise,’ but a small fraction contain ‘signal.’ We measure risk by per-coordinate mean-squared error in recovering log(μi), and study minimax estimation over parameter spaces defined by…

无限维指数族贝叶斯密度估计的收敛速度AI 解读

We study the rate of convergence of posterior distributions in density estimation problems for log-densities in periodic Sobolev classes characterized by a smoothness parameter p. The posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under Hellinger loss if the posterior distribution achieves the optimal rate over certain uniformity classes. A prior on the density class of interest is induced by a prior…

适用于短程与长程依赖的频域经验似然AI 解读

This paper introduces a version of empirical likelihood based on the periodogram and spectral estimating equations. This formulation handles dependent data through a data transformation (i.e., a Fourier transform) and is developed in terms of the spectral distribution rather than a time domain probability distribution. The asymptotic properties of frequency domain empirical likelihood are studied for linear time processes exhibiting both short- and long-range dependence. The…

分数协整子空间的半参数估计AI 解读

We consider a common-components model for multivariate fractional cointegration, in which the $s\geq1$ components have different memory parameters. The cointegrating rank may exceed 1. We decompose the true cointegrating vectors into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters. We estimate each cointegrating subspace separately, using appropriate sets of eigenvectors of an…

渐近极小极大贝叶斯预测密度AI 解读

Given a random sample from a distribution with density function that depends on an unknown parameter θ, we are interested in accurately estimating the true parametric density function at a future observation from the same distribution. The asymptotic risk of Bayes predictive density estimates with Kullback–Leibler loss function D(fθ||{f̂})=∫fθ log(fθ/f̂) is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We seek…

A depth-based rank sum statistic for multivariate data introduced by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252–260] as an extension of the Wilcoxon rank sum statistic for univariate data has been used in multivariate rank tests in quality control and in experimental studies. Those applications, however, are based on a conjectured limiting distribution, provided by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252–260]. The present paper proves the conjecture…

高斯局部平稳过程的非参数准最大似然估计AI 解读

This paper deals with nonparametric maximum likelihood estimation for Gaussian locally stationary processes. Our nonparametric MLE is constructed by minimizing a frequency domain likelihood over a class of functions. The asymptotic behavior of the resulting estimator is studied. The results depend on the richness of the class of functions. Both sieve estimation and global estimation are considered. Our results apply, in particular, to estimation under shape constraints. As…

广义部分线性模型中的稳健估计AI 解读

In this paper, we introduce a family of robust estimates for the parametric and nonparametric components under a generalized partially linear model, where the data are modeled by yi|(xi, ti)∼F(⋅, μi) with μi=H(η(ti)+xiTβ), for some known distribution function F and link function H. It is shown that the estimates of β are root-n consistent and asymptotically normal. Through a Monte Carlo study, the performance of these estimators is compared with that of the classical ones.

形状矩阵的半参数有效秩推断（二）：最优 R 估计AI 解读

A class of R-estimators based on the concepts of multivariate signed ranks and the optimal rank-based tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006) 2707–2756] is proposed for the estimation of the shape matrix of an elliptical distribution. These R-estimators are root-n consistent under any radial density g, without any moment assumptions, and semiparametrically efficient at some prespecified density f. When based on normal scores, they are uniformly…

形状参数的半参数有效秩推断（一）：最优球形性秩检验AI 解读

We propose a class of rank-based procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale and radial density) has some fixed value V0; this includes, for V0={I}k, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations and reflections with respect to the estimated center of symmetry. They are valid…

线性与非线性自回归模型的有效预测AI 解读

Conditional expectations given past observations in stationary time series are usually estimated directly by kernel estimators, or by plugging in kernel estimators for transition densities. We show that, for linear and nonlinear autoregressive models driven by independent innovations, appropriate smoothed and weighted von Mises statistics of residuals estimate conditional expectations at better parametric rates and are asymptotically efficient. The proof is based on a…

条件异方差时间序列的准极大似然估计：随机递推方程方法AI 解读

This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt=σtZt, where the unobservable volatility σt is a parametric function of (Xt−1, …, Xt−p, σt−1, …, σt−q) for some p, q≥0, and (Zt) is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are…

非参数二元回归中高斯过程先验的后验一致性AI 解读

Consider binary observations whose response probability is an unknown smooth function of a set of covariates. Suppose that a prior on the response probability function is induced by a Gaussian process mapped to the unit interval through a link function. In this paper we study consistency of the resulting posterior distribution. If the covariance kernel has derivatives up to a desired order and the bandwidth parameter of the kernel is allowed to take arbitrarily small values,…