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一类对称性 Sobolev 检验的检测阈值与渐近功效AI 解读

We consider a broad class of symmetry hypothesis testing problems that includes the problems of testing uniformity or rotational symmetry on the hypersphere Sd−1, as well as the problem of testing sphericity in Rd. For this class, we study the null and non-null behaviors of Sobolev tests, with emphasis on their consistency rates and corresponding asymptotic powers. Our main results show that: (i) Sobolev tests exhibit a detection threshold that depends not only on the…

弱可识别条件下主成分分析的秩检验AI 解读

In a triangular array framework where n observations are randomly sampled from a p-dimensional elliptical distribution with shape matrix Vn, we consider the problem of testing the null hypothesis H0:θ=θ0 against the alternative hypothesis H1:θ≠θ0, where θ is the (fixed) leading unit eigenvector of Vn and θ0 is a given unit p-vector. The dependence of the shape matrix on the sample size allows us to consider challenging asymptotic scenarios in which the parameter of interest…

单位超球面上的空间分位数AI 解读

We propose a concept of quantiles for probability measures on the unit hypersphere Sd−1 of Rd. The innermost quantile is the Fréchet median, that is, the L1-analog of the Fréchet mean. The proposed quantiles μα,um are directional in nature: they are indexed by a scalar order α∈[0,1] and a unit vector u in the tangent space TmSd−1 to Sd−1 at m. To ensure computability in any dimension d, our quantiles are essentially obtained by considering the Euclidean (Chaudhuri (J. Amer.…

预先多重检验估计及其在多样本协方差估计中的应用AI 解读

Multisample covariance estimation—that is, estimation of the covariance matrices associated with k distinct populations—is a classical problem in multivariate statistics. A common solution is to base estimation on the outcome of a test that these covariance matrices show some given pattern. Such a preliminary test may, for example, investigate whether or not the various covariance matrices are equal to each other (test of homogeneity), or whether or not they have common…

Lp 损失下多元位置的仿射等变推断AI 解读

We consider the fundamental problem of estimating the location of a d-variate probability measure under an Lp loss function. The naive estimator, that minimizes the usual empirical Lp risk, has a known asymptotic behavior but suffers from several deficiencies for p≠2, the most important one being the lack of equivariance under general affine transformations. In this work, we introduce a collection of Lp location estimators μˆnp,ℓ that minimize the size of suitable…

针对新型超球面分布备择的旋转对称性最优检验AI 解读

Motivated by the central role played by rotationally symmetric distributions in directional statistics, we consider the problem of testing rotational symmetry on the hypersphere. We adopt a semiparametric approach and tackle problems where the location of the symmetry axis is either specified or unspecified. For each problem, we define two tests and study their asymptotic properties under very mild conditions. We introduce two new classes of directional distributions that…

高集中度下球面位置参数的推断AI 解读

Motivated by the fact that circular or spherical data are often much concentrated around a location $\pmb{\theta }$, we consider inference about $\pmb{\theta }$ under high concentration asymptotic scenarios for which the probability of any fixed spherical cap centered at $\pmb{\theta }$ converges to one as the sample size $n$ diverges to infinity. Rather than restricting to Fisher–von Mises–Langevin distributions, we consider a much broader, semiparametric, class of…

弱可识别性下的主成分方向检验AI 解读

We consider the problem of testing, on the basis of a $p$-variate Gaussian random sample, the null hypothesis $\mathcal{H}_{0}:\boldsymbol{\theta}_{1}=\boldsymbol{\theta}_{1}^{0}$ against the alternative $\mathcal{H}_{1}:\boldsymbol{\theta}_{1}\neq \boldsymbol{\theta}_{1}^{0}$, where $\boldsymbol{\theta}_{1}$ is the “first” eigenvector of the underlying covariance matrix and $\boldsymbol{\theta}_{1}^{0}$ is a fixed unit $p$-vector. In the classical setup where eigenvalues $\lambda_{1}>\lambda_{2}\geq \cdots \geq \lambda_{p}$…

散布、精度与形状矩阵的半空间深度AI 解读

We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept is similar to those from Chen, Gao and Ren [Robust covariance and scatter matrix estimation under Huber’s contamination model (2018)] and Zhang [J. Multivariate Anal. 82 (2002) 134–165]. Rather than focusing, as in these earlier works, on deepest scatter matrices, we thoroughly investigate the properties of the proposed depth and of the corresponding depth…

针对单调旋转对称备择的高维球面均匀性检验AI 解读

We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in nonnull issues. We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for fixed modal location $\mathbf{{\theta}}$ and allows to derive locally asymptotically most powerful tests under specified $\mathbf{{\theta}}$. The second one, that addresses the Fisher–von Mises–Langevin (FvML) case, relates…

弱方向信号模态的推断：奇点附近假设检验的 Le Cam 视角AI 解读

We revisit, in an original and challenging perspective, the problem of testing the null hypothesis that the mode of a directional signal is equal to a given value. Motivated by a real data example where the signal is weak, we consider this problem under asymptotic scenarios for which the signal strength goes to zero at an arbitrary rate $\eta_{n}$. Both under the null and the alternative, we focus on rotationally symmetric distributions. We show that, while they are…

主成分与共同主成分的高效秩估计AI 解读

We propose rank-based estimators of principal components, both in the one-sample and, under the assumption of common principal components, in the m-sample cases. Those estimators are obtained via a rank-based version of Le Cam’s one-step method, combined with an estimation of cross-information quantities. Under arbitrary elliptical distributions with, in the m-sample case, possibly heterogeneous radial densities, those R-estimators remain root-n consistent and asymptotically…

从数据深度到局部深度：聚焦中心性AI 解读

Aiming at analyzing multimodal or nonconvexly supported distributions through data depth, we introduce a local extension of depth. Our construction is obtained by conditioning the distribution to appropriate depth-based neighborhoods and has the advantages, among others, of maintaining affine-invariance and applying to all depths in a generic way. Most importantly, unlike their competitors, which (for extreme localization) rather measure probability mass, the resulting local…

摘要暂缺

对称独立成分模型中基于符号秩的半参数有效推断AI 解读

We consider semiparametric location-scatter models for which the p-variate observation is obtained as X = ΛZ + μ, where μ is a p-vector, Λ is a full-rank p × p matrix and the (unobserved) random p-vector Z has marginals that are centered and mutually independent but are otherwise unspecified. As in blind source separation and independent component analysis (ICA), the parameter of interest throughout the paper is Λ. On the basis of n i.i.d. copies of X, we develop, under a…

主成分的最优秩检验AI 解读

This paper provides parametric and rank-based optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families. The parametric tests extend the Gaussian likelihood ratio tests of Anderson (1963) and their pseudo-Gaussian robustifications by Davis (1977) and Tyler (1981, 1983). The rank-based tests address a much broader class of problems, where covariance matrices need not exist and principal components are associated with more general…

A new multivariate concept of quantile, based on a directional version of Koenker and Bassett’s traditional regression quantiles, is introduced for multivariate location and multiple-output regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to…

多变量随机性的游程检验AI 解读

This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999), and (ii) an original concept of matrix-valued runs. The resulting runs tests themselves exist in various versions, one of which is a function of the number of data-based hyperplanes…

散布矩阵同质性的最优秩检验AI 解读

We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavy-tailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain…

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…

椭圆分布 VARMA 模型适合性的秩最优检验AI 解读

We are deriving optimal rank-based tests for the adequacy of a vector autoregressive-moving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudo-Mahalanobis distances and on normed residuals computed from Tyler’s [Ann. Statist. 15 (1987) 234–251] scatter matrix; they generalize the univariate signed rank procedures proposed by Hallin and Puri [J. Multivariate Anal. 39 (1991) 1–29]. Two types of optimality…

We propose a family of tests, based on Randles' (1989) concept of interdirections and the ranks of pseudo-Mahalanobis distances computed with respect to a multivariate M-estimator of scatter due to Tyler (1987), for the multivariate one-sample problem under elliptical symmetry. These tests, which generalize the univariate signed-rank tests, are affine-invariant. Depending on the score function considered (van der Waerden, Laplace,...), they allow for locally asymptotically…