Related papers: Semicircle Law for Tyler's M-Estimator of Scatter

This paper studies the limiting behavior of Tyler's M-estimator for the scatter matrix, in the regime that the number of samples $n$ and their dimension $p$ both go to infinity, and $p/n$ converges to a constant $y$ with $0<y<1$. We prove…

This survey provides a self-contained account of $M$-estimation of multivariate scatter. In particular, we present new proofs for existence of the underlying $M$-functionals and discuss their weak continuity and differentiability. This is…

In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability is $O(n^{-1/2})$…

This paper analyzes the performance of Tyler's M-estimator of the scatter matrix in elliptical populations. We focus on the non-asymptotic setting and derive the estimation error bounds depending on the number of samples n and the dimension…

In this paper, {we propose an alternative proof for the uniqueness} of Maronna's $M$-estimator of scatter (Maronna, 1976) for $N$ vector observations $\mathbf y_1,...,\mathbf y_N\in\mathbb R^m$ under a mild constraint of linear independence…

We introduce kernel estimators for the semicircle law. In this first part of our general theory on the estimators, we prove the consistency and conduct simulation study to show the performance of the estimators. We also point out that…

Estimating the shape of an elliptical distribution is a fundamental problem in statistics. One estimator for the shape matrix, Tyler's M-estimator, has been shown to have many appealing asymptotic properties. It performs well in numerical…

This paper considers the regularized Tyler's scatter estimator for elliptical distributions, which has received considerable attention recently. Various types of shrinkage Tyler's estimators have been proposed in the literature and proved…

The celebrated Trotter approximation theorem provides a sufficient condition for the convergence of a sequence of operator semigroups in terms of the corresponding sequence of infinitesimal generators. There exist a few results on the rate…

The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which…

We present new algorithms for $M$-estimators of multivariate scatter and location and for symmetrized $M$-estimators of multivariate scatter. The new algorithms are considerably faster than currently used fixed-point and related algorithms.…

We consider random Hermitian matrices with independent upper triangular entries. Wigner's semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to the semicircle distribution. We…

Maronna's and Tyler's $M$-estimators are among the most widely used robust estimators for scatter matrices. However, when the dimension of observations is relatively high, their performance can substantially deteriorate in certain…

Scattering of starlight by dust, molecules and electrons in spiral galaxies will produce a modification of the direct intensity and a polarization in the observed light. We treat the case where the distribution of scatterers can be…

We investigate the spectral distribution of random matrix ensembles with correlated entries. We consider symmetric matrices with real valued entries and stochastically independent diagonals. Along the diagonals the entries may be…

We consider in this paper the semiparametric mixture of two distributions equal up to a shift parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. In…

The traditional class of elliptical distributions is extended to allow for asymmetries. A completely robust dispersion matrix estimator (the `spectral estimator') for the new class of `generalized elliptical distributions' is presented. It…

Let $X$ be a matrix sampled uniformly from the set of doubly stochastic matrices of size $n\times n$. We show that the empirical spectral distribution of the normalized matrix $\sqrt{n}(X-{\mathbf {E}}X)$ converges almost surely to the…

We consider the nonlinear Schr{\"o}dinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a com-plete scattering theory is available, showing that both the potential and…

The theory of elastic light scattering by semiconductor quantum dots is suggested. The semiclassical method, applying retarded potentials to avoid the problem of bounder conditions for electric and magnetic field, is used. The exact results…