The Frechet mean is a useful description of location for a probability distribution on a metric space that is not necessarily a vector space. This article considers simultaneous estimation of multiple Frechet means from a decision-theoretic perspective, and in particular, the extent to which the unbiased estimator of a Frechet mean can be dominated by a generalization of the James-Stein shrinkage estimator. It is shown that if the metric space satisfies a non-positive curvature condition, then this generalized James-Stein estimator asymptotically dominates the unbiased estimator as the dimension of the space grows. These results hold for a large class of distributions on a variety of spaces - including Hilbert spaces - and therefore partially extend known results on the applicability of the James-Stein estimator to non-normal distributions on Euclidean spaces. Simulation studies on metric trees and symmetric-positive-definite matrices are presented, numerically demonstrating the efficacy of this generalized James-Stein estimator.
Data in non-Euclidean spaces are commonly encountered in many fields of Science and Engineering. For instance, in Robotics, attitude sensors capture orientation which is an element of a Lie group. In the recent past, several researchers have reported methods that take into account the geometry of Lie Groups in designing parameter estimation algorithms in nonlinear spaces. Maximum likelihood estimators (MLE) are quite commonly used for such tasks and it is well known in the field of statistics that Stein's shrinkage estimators dominate the MLE in a mean-squared sense assuming the observations a…
Transformed Fréchet Means for Robust Estimation in Hadamard Spaces
Technical University of Munich（慕尼黑工业大学）；Munich Climate Center（慕尼黑气候中心）；TUM School of Engineering and Design, Department of Aerospace and Geodesy, Earth System Modelling Group（慕尼黑工业大学工程与设计学院，航空航天与大地测量系，地球系统建模组）；Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）；Artificial Intelligence Group（人工智能小组）
Uniform Convergence of Generalized Conditional Fréchet Means with Applications to Weighted Fréchet Aggregation and Exceedance Set Estimation
Research School of Finance, Actuarial Studies and Statistics, Australian National University（澳大利亚国立大学金融、精算与统计研究学院）；Department of Actuarial Studies and Business Analytics, Macquarie University（麦考瑞大学精算与商业分析系）
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Evry（巴黎萨克雷大学，法国国家科学研究中心，埃夫里大学，埃夫里数学与建模实验室）；Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189, CRIStAL（里尔大学，法国国家信息与自动化研究所，法国国家科学研究中心，里尔中央理工学院，CRIStAL实验室）；LPSM of Sorbonne University（索邦大学LPSM）
Omnibus CLTs for Fréchet means and nonparametric inference on non-Euclidean spaces
得到了样本Fréchet平均的两个中心极限定理，这两个定理对于非欧几里得空间上的非参数推断都有重要意义.第一个，定理2.2，包括和改进最早的CLTs上Fréchet的手段和扩大范围的方法超越流形，以不同的新的非欧几里德数据，包括那些在某些分层空间是重要的系统发育树的研究。它不要求底层分布$Q$具有密度，并且适用于内在和外在分析。第二个定理，定理3.3，集中在维数为$d>2$的黎曼流形上的内在平均，并通过提供一个广泛的CLT而没有任何早期的限制性支持假设来开辟新的领域。它对$Q$的某种平滑密度做出了统计上合理的假设。维数$d=2$的排除情形被证明是一个谜，尽管第一个定理在这种情形下也在支撑限制下提供了一个CLT。定理3.3直接适用于球体$S^d$、$d>2$，它们在轴向空间和基于界标的图像分析的应用中也是相当重要的，因为这些空间是球体在$S^d$的等距李群$\mathcal G $下的等价物。
Differential privacy has become crucial in the real-world deployment of statistical and machine learning algorithms with rigorous privacy guarantees. The earliest statistical queries, for which differential privacy mechanisms have been developed, were for the release of the sample mean. In Geometric Statistics, the sample Fréchet mean represents one of the most fundamental statistical summaries, as it generalizes the sample mean for data belonging to nonlinear manifolds. In that spirit, the only geometric statistical query for which a differential privacy mechanism has been developed, so far, …
In the past decades, the central limit theorem (CLT) has been generalized to non-Euclidean data spaces. Some years ago, it was found that for some random variables on the circle, the sample Fréchet mean fluctuates around the population mean asymptotically at a scale $n^{-τ}$ with exponent $τ< 1/2$ with a non-normal distribution if the probability density at the antipodal point of the mean is $\frac{1}{2π}$. The author and his collaborator recently discovered that $τ= 1/6$ for some random variables on higher dimensional spheres. In this article we show that, even more surprisingly, the phenomen…
The Fréchet mean generalizes the concept of a mean to a metric space setting. In this work we consider equivariant estimation of Fréchet means for parametric models on metric spaces that are Riemannian manifolds. The geometry and symmetry of such a space is encoded by its isometry group. Estimators that are equivariant under the isometry group take into account the symmetry of the metric space. For some models there exists an optimal equivariant estimator, which necessarily will perform as well or better than other common equivariant estimators, such as the maximum likelihood estimator or the …
We prove a central limit theorem (CLT) for the Frechet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Frechet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Frechet mean lies outside the support of the population distribution. So far as we are aware, the CLT in the present paper is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approxima…
Second order Stein: SURE for SURE and other applications in high-dimensional inference
Stein公式指出，对于梯度可积的函数$f$，形式为$z^\top f(z) - \text{div} f(z)$的随机变量是均值零。这里，$\text{div} f$是函数$f$的散度，而$z$是标准法向量。本文的目的是提出一个二阶Stein公式来描述所有函数$f(z)$的平方可积梯度的随机变量的方差，并证明该公式在各种应用中的有用性。在高斯序列模型中，Stein公式的一个结果是Stein无偏风险估计（SURE），这是对未知均值的几乎任何估计量$\hatμ$的均方风险的无偏估计。二阶Stein公式的第一个应用是SURE本身的无偏风险估计（SURE for SURE）：无偏估计{提供}关于SURE与$\hatμ$的平方估计误差之间的平方距离的信息。SURE for SURE具有作为数据函数的简单形式，适用于具有平方可积梯度的所有$\hatμ$，例如Lasso和Elastic Net。除了SURE for SURE之外，还开发了以下应用：（1）当估计目标为均方误差时SURE风险的上界;（2）基于SURE的置信区域;（3）SURE调整估计所满足的Oracle不等式;（4）Lasso选择的模型大小方差的上界;（5）在Lasso和Elastic-Net中SURE的SURE显式表达式;（6）在线性模型中，一个通用的半参数方案，用于对未知$β$的低维投影的推断的可微初始估计量进行去偏，并在去偏…
The Stein paradox has played an influential role in the field of high dimensional statistics. This result warns that the sample mean, classically regarded as the "usual estimator", may be suboptimal in high dimensions. The development of the James-Stein estimator, that addresses this paradox, has by now inspired a large literature on the theme of "shrinkage" in statistics. In this direction, we develop a James-Stein type estimator for the first principal component of a high dimension and low sample size data set. This estimator shrinks the usual estimator, an eigenvector of a sample covariance…
MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；The Institute of Statistical Mathematics（统计数学研究所）；University of Cambridge（剑桥大学）；University College London（伦敦大学学院）
Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparametric Fréchet regression, which refers to a regression setting where a response residing in a metric space is paired with a scalar predictor and the target is a conditional Fréchet mean. Specifically, we seek to approximate an unknown metric-space valued function…
This article discusses estimation of a multivariate normal mean based on heteroscedastic observations. Under heteroscedasticity, estimators shrinking more on the coordinates with larger variances, seem desirable. Although they are not necessarily minimax in the ordinary sense, we show that such James-Stein type estimators can be ensemble minimax, minimax with respect to the ensemble risk, related to empirical Bayes perspective of Efron and Morris.
