Robert Serfling, Yijun Zuo
Discussion of "Multivariate quantiles and multiple-output regression quantiles: From $L_1$ optimization to halfspace depth" by M. Hallin, D. Paindaveine and M. Siman [arXiv:1002.4486]
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Discussion of "Multivariate quantiles and multiple-output regression quantiles: From $L_1$ optimization to halfspace depth" by M. Hallin, D. Paindaveine and M. Siman [arXiv:1002.4486]
Discussion of "Multivariate quantiles and multiple-output regression quantiles: From $L_1$ optimization to halfspace depth" by M. Hallin, D. Paindaveine and M. Siman [arXiv:1002.4486]
Rejoinder to "Multivariate quantiles and multiple-output regression quantiles: From $L_1$ optimization to halfspace depth" by M. Hallin, D. Paindaveine and M. Siman [arXiv:1002.4486]
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 coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow f…
Despite the renewed interest in the Newey and Powell (1987) concept of expectiles in fields such as econometrics, risk management, and extreme value theory, expectile regression---or, more generally, M-quantile regression---unfortunately remains limited to single-output problems. To improve on this, we introduce hyperplane-valued multivariate M-quantiles that show strong advantages, for instance in terms of equivariance, over the various point-valued multivariate M-quantiles available in the literature. Like their competitors, our multivariate M-quantiles are directional in nature and provide …
Extremal behaviour and convergence rates for sample--based geometric quantiles and half space depths
该研究分析基于样本的几何分位数与半空间深度的极值行为，建立收敛速率与基础分布尾部行为的关联，揭示样本量与极值参数的复杂作用，填补总体版本相关认知空白。
Tukey's halfspace median ($\HM$), servicing as the {multivariate} counterpart of the univariate median, has been introduced and extensively studied in the literature. It is supposed and expected to preserve robustness property (the most outstanding property) of the univariate median. One of prevalent quantitative assessments of robustness is finite sample breakdown point (FSBP). Indeed, the FSBP of many multivariate medians have been identified, except for the most prevailing one---the Tukey's halfspace median. This paper presents a precise result on FSBP for Tukey's halfspace median. The resu…
The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to higher-dimensional spaces. The smallest non-empty trimmed region, coined the halfspace median of a measure, generalizes the median. We focus on the (inverse) ray basis theorem for the halfspace depth, a crucial theoretical result that characterizes the halfspace median by a covering property. First, a novel elementary proof of that statement is provided, under …
We give examples of different multivariate probability distributions whose halfspace depths coincide at all points of the sample space.
Theoretical properties of angular halfspace depth
本文全面探讨了角半空间深度（ahD）的理论性质，证明其满足统计深度函数的理想性质，并推导了方向中位数集合与中心区域的一致连续性/一致性结果。
Set-valued quantiles for multivariate distributions with respect to a general convex cone are introduced which are based on a family of (univariate) distribution functions rather than on the joint distribution function. It is shown that these quantiles enjoy basically all the properties of univariate quantile functions. Relationships to families of univariate quantile functions and to depth functions are discussed. Finally, a corresponding Value at Risk for multivariate random variables as well as stochastic orders are introduced via the set-valued approach.
The univariate quantile-quantile (Q-Q) plot is a well-known graphical tool for examining whether two data sets are generated from the same distribution or not. It is also used to determine how well a specified probability distribution fits a given sample. In this article, we develop and study a multivariate version of the Q-Q plot based on the spatial quantile. The usefulness of the proposed graphical device is illustrated on different real and simulated data, some of which have fairly large dimensions. We also develop certain statistical tests that are related to the proposed multivariate Q-Q…
Location and scatter halfspace median under α-symmetric distributions
本文研究了α对称分布下位置和散射半空间中位数的估计性能，提出改进的散射半空间深度以适应更广泛分布，并推导了相关上界。
Extreme Geometric Quantiles Under Minimal Assumptions, with a Connection to Tukey Depth
本文在最小假设下研究了极端几何四分位数的性质，并揭示了其与Tukey深度之间的新联系。
In this paper, we consider Bayesian inference on a class of multivariate median and the multivariate quantile functionals of a joint distribution using a Dirichlet process prior. Since, unlike univariate quantiles, the exact posterior distribution of multivariate median and multivariate quantiles are not obtainable explicitly, we study these distributions asymptotically. We derive a Bernstein-von Mises theorem for the multivariate $\ell_1$-median with respect to general $\ell_p$-norm, which in particular shows that its posterior concentrates around its true value at $n^{-1/2}$-rate and its cre…
Multivariate analysis-of-variance (MANOVA) is a well established tool to examine multivariate endpoints. While classical approaches depend on restrictive assumptions like normality and homogeneity, there is a recent trend to more general and flexible proce dures. In this paper, we proceed on this path, but do not follow the typical mean-focused perspective. Instead we consider general quantiles, in particular the median, for a more robust multivariate analysis. The resulting methodology is applicable for all kind of factorial designs and shown to be asymptotically valid. Our theoretical result…
Univariate L-moments are expressed as projections of the quantile function onto an orthogonal basis of polynomials in $L_2([0;1],\mathbb{R})$. We present multivariate versions of L-moments expressed as collections of orthogonal projections of a multivariate quantile function on a basis of multivariate polynomials in $L_2([0;1]^d,\mathbb{R})$. We propose to consider quantile functions defined as transport from the uniform distribution on $[0;1]^d$ onto the distribution of interest. In particular, we present the quantiles defined by the transport of Rosenblatt and the optimal transport and the p…
We define the quantile set of order $α\in \left[ 1/2,1\right) $ associated to a law $P$ on $\mathbb{R}^{d}$ to be the collection of its directional quantiles seen from an observer $O\in \mathbb{R}^{d}$. Under minimal assumptions these star-shaped sets are closed surfaces, continuous in $(O,α)$ and the collection of empirical quantile surfaces is uniformly consistent.\ Under mild assumptions -- no density or symmetry is required for $P$ -- our uniform central limit theorem reveals the correlations between quantile points and a non asymptotic Gaussian approximation provides joint confident enlar…
For any given partial order in a $d$-dimensional euclidean space, under mild regularity assumptions, we show that the intersection of closed (generalized) intervals containing more than 1/2 of the probability mass, is a non-empty compact interval. This property is shared with common intervals on real line, where the intersection is the median set of the underlying probability distribution. So obtained multivariate medians with respect to a partial order, can be observed as special cases of centers of distribution in the sense of type D depth functions introduced by Y. Zuo and R. Serfling, {\em…
Finite Representation of Quantile Sets for Multivariate Data via Vector Linear Programming
本文指出可通过求解向量线性规划获得多元经验分位数，为计算Tukey深度区域和更一般的锥分位数集提供了新方法。
Geometric quantile-based measures of multivariate distributional characteristics
本文提出了基于几何分位数的多元分布特征（离散度、偏度、峰度及球面不对称性）新度量，并通过理论证明和实验验证了其合理性与置信区域计算的准确性。
In this paper we study multivariate ranks and quantiles, defined using the theory of optimal transport, and build on the work of Chernozhukov et al.(2017) and Hallin et al.(2021). We study the characterization, computation and properties of the multivariate rank and quantile functions and their empirical counterparts. We derive the uniform consistency of these empirical estimates to their population versions, under certain assumptions. In fact, we prove a Glivenko-Cantelli type theorem that shows the asymptotic stability of the empirical rank map in any direction. Under mild structural assumpt…
Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours
本文比较了多元分位数扩展中的两种主要概念——geometric quantiles与基于measure transportation的center-outward quantiles，分析了它们的主要理论性质与数值差异。
A new quantile regression concept, based on a directional version of Koenker and Bassett's traditional single-output one, has been introduced in [Ann. Statist. (2010) 38 635-669] for multiple-output location/linear regression problems. The polyhedral contours provided by the empirical counterpart of that concept, however, cannot adapt to unknown nonlinear and/or heteroskedastic dependencies. This paper therefore introduces local constant and local linear (actually, bilinear) versions of those contours, which both allow to asymptotically recover the conditional halfspace depth contours that com…
In multivariate extreme value theory (MEVT), the focus is on analysis outside of the observable sampling zone, which implies that the region of interest is associated to high risk levels. This work provides tools to include directional notions into the MEVT, giving the opportunity to characterize the recently introduced directional multivariate quantiles (DMQ) at high levels. Then, an out-sample estimation method for these quantiles is given. A bootstrap procedure carries out the estimation of the tuning parameter in this multivariate framework and helps with the estimation of the DMQ. Asympto…
Quantiles for multivariate distribution: a short survey of population concepts
本文综述了多元分布中quantiles的总体概念，指出一维定义依赖实数序而高维缺乏典范序，总结了多种扩展定义并更新了Serfling的经典综述。
We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution of interest on $R^d$ and a reference distribution on the $d$-dimensional unit ball. The new depth concept, called Monge-Kantorovich depth, specializes to halfspace depth in the case of spherical distributions, but, for more general distributions, differs from the latter in the ability for its contours to account for non convex features of the distribution of interest. We propose empirical counterparts to the population versions of those Monge-Kant…
Unified unconditional regression for multivariate quantiles, M-quantiles and expectiles
该研究提出基于多维Huber函数的统一无条件回归方法，可对多元因变量的三类分位数建模，拓展了已有重中心化影响函数回归框架，证明了估计量渐近性质，给出调参选择方法，并通过模拟和真实数据验证了有效性。
Quantile-stratified sampling for multivariate normal simulations and other multivariate distributions
本文提出将分位数分层抽样扩展至多元分布模拟的方法，该方法生成的样本空间填充性与覆盖性优于IID抽样，通过有序对数密度值图验证了其覆盖性能。
Bayesian and other likelihood-based methods require specification of a statistical model and may not be fully satisfactory for inference on quantities, such as quantiles, that are not naturally defined as model parameters. In this paper, we construct a direct and model-free Gibbs posterior distribution for multivariate quantiles. Being model-free means that inferences drawn from the Gibbs posterior are not subject to model misspecification bias, and being direct means that no priors for or marginalization over nuisance parameters are required. We show here that the Gibbs posterior enjoys a roo…
Nonparametric Vector Quantile Autoregression
针对现有分位数自回归方法受限于单变量时间序列的问题，本文基于测度传输型多元分位数概念，提出非参数向量分位数自回归方法，可实现多元时间序列的全分布预测。
The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the sample points, naturally breaks ties in the associated depth-based ordering, and introduces a depth-like function for points outside the convex hull of the support of the probability measure. The illumination is, in a certain sense, dual to the halfspace depth mapping, and shares the majority of its beneficial properties. It is affine invariant, robust, uniformly consistent, and aligns well with common prob…
Monge-Kantorovich superquantiles and expected shortfalls with applications to multivariate risk measurements
本文基于最优传输理论提出Monge-Kantorovich超分位数与预期短缺函数，将风险价值和条件风险价值扩展至多元空间，用于多元风险度量，并证明其能刻画随机向量及其依分布收敛性。
We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain confidence corridors for the regression function in the classical mean regression. In order to deal with th…
Generative Modelling via Quantile Regression
本文将条件生成建模与分位数回归相联系，提出损失函数并推导极小极大收敛速率，通过非参数回归子问题建立下界，还讨论了多元分布数据生成扩展。
Regularized Halfspace Depth for Functional Data
针对函数型数据，提出一种通过正则化避免退化的半空间深度，具备理想理论性质，可排序数据并检测形状与幅度离群点，经数值与真实数据验证有效。
Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur's representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that \[\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^nϕ\bigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}\bigr)-\barγ\Biggr)=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_{n,i}+\mathrm{o}_P(1)\] as $n\rightarrow\inft…
Discriminating Tail Behavior Using Halfspace Depths: Population and Empirical Perspectives
本文研究半空间深度的经验版本，建立收敛速率与底层分布尾部行为的联系，并基于加权经验过程提出评估多元分布尾部行为的方法。
This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of partial quantiles, which are based on a given partial order. We establish that partial quantiles are equivariant under order-preserving transformations of the data, robust to outliers, characterize the probability distribution if the partial order is sufficiently rich, generalize the concept of efficient frontier, and can measure dispersion from the partial order perspective. We also study several statistical aspects of partial quantiles. We provide estimators, associated rates of convergenc…
Extending to dimension 2 and higher the dual univariate concepts of ranks and quantiles has remained an open problem for more than half a century. Based on measure transportation results, a solution has been proposed recently under the name center-outward ranks and quantiles which, contrary to previous proposals, enjoys all the properties that make univariate ranks a successful tool for statistical inference. Just as their univariate counterparts (to which they reduce in dimension one), center-outward ranks allow for the construction of distribution-free and asymptotically efficient tests for …
Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persis…
In this article, we propose a penalized high dimensional semiparametric model average quantile prediction approach that is robust for forecasting the conditional quantile of the response. We consider a two-step estimation procedure. In the first step, we use a local linear regression approach to estimate the individual marginal quantile functions, and approximate the conditional quantile of the response by an affine combination of one-dimensional marginal quantile regression functions. In the second step, based on the nonparametric kernel estimates of the marginal quantile regression functions…
Comparing Multivariate Distributions: A Novel Approach Using Optimal Transport-based Plots
本文提出利用最优传输和熵正则化最优传输构建多元 Q-Q 图及势函数图，以捕捉高维数据依赖关系和分布差异，并给出相应检验统计量。
Projection Diagnostics for Directional Asymmetry and Tail-Ratio Departure in Multivariate Data
提出基于投影的诊断方法，通过方向偏度与分位数尾部比率将数据分类为四种模式，避免高阶矩的不稳定性，并建立理论性质。
Bias robustness of depth estimators in multivariate settings
本文研究了多变量环境下深度估计器的偏置鲁棒性，分析了Tukey中位数、基于深度的散射矩阵和多元回归估计器的统计收敛性和鲁棒性。
We study quantile trend filtering, a recently proposed method for nonparametric quantile regression with the goal of generalizing existing risk bounds known for the usual trend filtering estimators which perform mean regression. We study both the penalized and the constrained version (of order $r \geq 1$) of univariate quantile trend filtering. Our results show that both the constrained and the penalized version (of order $r \geq 1$) attain the minimax rate up to log factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded variation signal…
On the use of the M-quantiles for outlier detection in multivariate data
本文研究了用于异常值检测的M-quantiles，K-transform，证明了在奇数维度d>=3中其与密度成正比的是多重拉普拉斯算子而非梯度，，揭示了该机制在异常值检测中的存在放大效应缺陷。
Based on the novel concept of multivariate center-outward quantiles introduced recently in Chernozhukov et al. (2017) and Hallin et al. (2021), we are considering the problem of nonparametric multiple-output quantile regression. Our approach defines nested conditional center-outward quantile regression contours and regions with given conditional probability content irrespective of the underlying distribution; their graphs constitute nested center-outward quantile regression tubes. Empirical counterparts of these concepts are constructed, yielding interpretable empirical regions and contours wh…
Estimating Systemic Risk within Financial Networks: A Two-Step Nonparametric Method
本文针对CoVaR估计中多元分位数条件不可观测的难题，提出基于蒙特卡洛模拟数据的两步非参数方法，先以次序统计量估计多元分位数，再用核方法估计条件分位数，并证明其一致性与渐近正态性，数值实验验证了方法的有效性。
Joint Quantile Shrinkage: A State-Space Approach toward Non-Crossing Bayesian Quantile Models
针对分位数回归模型中拟合条件分位数交叉的普遍问题，提出一种贝叶斯建模框架，通过带分位数适应性的融合收缩先验联合估计多个分位数，形成分位数变参数模型，在参数恢复和预测性能上更优。
We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Ri…
We examine the problem of computing multivariate scenarios sets for skewed distributions. Our interest is motivated by the potential use of such sets in the "stress testing" of insurance companies and banks whose solvency is dependent on changes in a set of financial "risk factors". We define multivariate scenario sets based on the notion of half-space depth (HD) and also introduce the notion of expectile depth (ED) where half-spaces are defined by expectiles rather than quantiles. We then use the HD and ED functions to define convex scenario sets that generalize the concepts of quantile and e…
We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that - in contrast to the nonparametric approach based on estimation of $L^2$-distances - the new…
Bayesian joint quantile autoregression
本文提出完全贝叶斯的联合分位数自回归（QAR）框架，扩展 Koenker-Xiao 与 Tokdar-Kadane 的方法，给出 QAR(p)、多元 QAR(1) 和空间 QAR(1) 模型，并用模拟和西班牙阿拉贡温度数据验证。
On the breakdown point of transport-based quantiles
本文推导了基于最优传输的多元分位数的崩溃点，证明了传输中位数及特定阶数的传输深度等值线点分别具有1/2和τ的崩溃点，表明其与一元对应物具有相同的稳健性。
Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects. In this paper, we consider statistical inference for quantile regression with large-scale data in the "increasing dimension" regime. We provide a comprehensive and in-depth analysis of a convolution-type smoothing approach that achieves adequate approximation to computation and inference for quantile regression. This method, which we refer to as {\it{conquer}}, turns the non-differentiable quantile loss function into a twice-differ…
A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this pur…
Differentially private multivariate medians
本文系统研究了基于深度的多元中位数在差分隐私多元位置估计中的应用，开发了近乎极小最优的有限样本性能保证，并在高达100维的高斯污染模型下验证了其优于重尾估计代价的结论。
Bivariate vine copula based regression, bivariate level and quantile curves
本文利用基于vine copula的二元回归模型的水平曲线构建二元条件分位数，提出一种对称处理双响应变量的图结构模型，有效避免了传统回归缺陷，并在天气数据中验证了联合建模的优势。
Polar Depth for Potentially Heavy-Tailed Data
提出极坐标深度函数用于分析多元重尾分布的极端值，证明其在大阈值下收敛到极限分布，并应用于异常检测。
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