Related papers: The Stein Effect for Frechet Means

Fr\'echet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fr\'echet means can be estimated from independent and identically distributed samples and uncover a fundamental…

A mean function in reproducing kernel Hilbert space, or a kernel mean, is an important part of many applications ranging from kernel principal component analysis to Hilbert-space embedding of distributions. Given finite samples, an…

We consider the problem of estimating the Fr\'echet and conditional Fr\'echet mean from data taking values in separable metric spaces. Unlike Euclidean spaces, where well-established methods are available, there is no practical estimator…

This paper reviews advances in Stein-type shrinkage estimation for spherically symmetric distributions. Some emphasis is placed on developing intuition as to why shrinkage should work in location problems whether the underlying population…

This paper presents a novel approach to constructing estimators that dominate the classical James-Stein estimator under the quadratic loss for multivariate normal means. Building on Stein's risk representation, we introduce a new sufficient…

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability…

Estimating the mean of a random vector from i.i.d. data has received considerable attention, and the optimal accuracy one may achieve with a given confidence is fairly well understood by now. When the data take values in more general metric…

The James-Stein estimator is a biased estimator -- for a finite number of samples its expected value is not the true mean. The maximum-likelihood estimator (MLE), is unbiased and asymptotically optimal. Yet, when estimating the mean of $3$…

Random forests are a statistical learning method widely used in many areas of scientific research because of its ability to learn complex relationships between input and output variables and also its capacity to handle high-dimensional…

We describe the use of the Frechet mean and variance in the Billera-Holmes-Vogtmann (BHV) treespace to summarize and explore the diversity of a set of phylogenetic trees. We show that the Frechet mean is comparable to other summary methods,…

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…

Shrinkage estimation usually reduces variance at the cost of bias. But when we care only about some parameters of a model, I show that we can reduce variance without incurring bias if we have additional information about the distribution of…

We introduce a location statistic for distributions on non-linear geometric spaces, the diffusion mean, serving as an extension and an alternative to the Fr\'echet mean. The diffusion mean arises as the generalization of Gaussian maximum…

Across many scientific disciplines, multiple observations are collected from the same experimental units, and in modern datasets these observations often arise as non-Euclidean random objects. In such settings, the incorporation of random…

The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in Euclidean space, but the extrinsic Euclidean mean often lies outside of treespace. Sturm showed that the intrinsic Frechet mean exists and is unique in treespace.…

Fr\'echet mean and variance provide a way of obtaining mean and variance for general metric space valued random variables and can be used for statistical analysis of data objects that lie in abstract spaces devoid of algebraic structure and…

Fr\'echet means are indispensable for nonparametric statistics on non-Euclidean spaces. For suitable random variables, in some sense, they "sense" topological and geometric structure. In particular, smeariness seems to indicate the presence…

The Reverse Stein Effect is identified and illustrated: A statistician who shrinks his/her data toward a point chosen without reliable knowledge about the underlying value of the parameter to be estimated but based instead upon the observed…

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…

It is well known, that Fr\'echet means on non-Euclidean spaces may exhibit nonstandard asymptotic rates depending on curvature. Even for distributions featuring standard asymptotic rates, there are non-Euclidean effects, altering finite…