Related papers: Custom-made Gauss quadrature for statisticians
Closed formulae for all Gaussian or optimal, 1-parameter quadrature rules in a compact interval [a, b] with non uniform, asymmetric subintervals, arbitrary number of nodes per subinterval for the spline classes $S_{2N, 0}$ and $S_{2N+1,…
In order to approximate the Riemann--Stieltjes integral $\int_a^b {f\left( t \right)dg\left( t \right)}$ by $2$--point Gaussian quadrature rule, we introduce the quadrature rule \begin{align*} \int_{ - 1}^1 {f\left( t \right)dg\left( t…
The study of hidden structures in data presents challenges in modern statistics and machine learning. We introduce the $\mathbf{gips}$ package in R, which identifies permutation subgroup symmetries in Gaussian vectors. $\mathbf{gips}$…
This article presents a rigorous analysis for efficient statistically accurate algorithms for solving the Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures.…
The quasi-maximum likelihood estimation is a commonly-used method for estimating GARCH parameters. However, such estimators are sensitive to outliers and their asymptotic normality is proved under the finite fourth moment assumption on the…
We introduce an estimator for the curvature of curves and surfaces by using finite sample points drawn from sampling a probability distribution that has support on the curve or surface. First we give an algorithm for estimation of the…
We formulate a reduced-order strategy for efficiently forecasting complex high-dimensional dynamical systems entirely based on data streams. The first step of our method involves reconstructing the dynamics in a reduced-order subspace of…
We describe a general technique that yields the first {\em Statistical Query lower bounds} for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning…
The goal of this paper is to show that a single robust estimator of the mean of a multivariate Gaussian distribution can enjoy five desirable properties. First, it is computationally tractable in the sense that it can be computed in a time…
Let $\mu$ be a positive Borel measure on the real line and let $L$ be the linear functional on univariate polynomials of bounded degree, defined as integration with respect to $\mu$. In 2020, Blekherman et al., the characterization of all…
Riemannian Gaussian distributions were initially introduced as basic building blocks for learning models which aim to capture the intrinsic structure of statistical populations of positive-definite matrices (here called covariance…
This paper investigates the approximation of Gaussian random variables in Banach spaces, focusing on the high-probability bounds for the approximation of Gaussian random variables using finitely many observations. We derive non-asymptotic…
Since Pearson [Philosophical Transactions of the Royal Society of London. A, 185 (1894), pp. 71-110] first applied the method of moments (MM) for modeling data as a mixture of one-dimensional Gaussians, moment-based estimation methods have…
We propose an estimator for the mean of a random vector in $\mathbb{R}^d$ that can be computed in time $O(n^4+n^2d)$ for $n$ i.i.d.~samples and that has error bounds matching the sub-Gaussian case. The only assumptions we make about the…
We prove a bound for the Wasserstein distance between vectors of smooth complex random variables and complex Gaussians in the framework of complex Markov diffusion generators. For the special case of chaotic eigenfunctions, this bound can…
The Lagrange-mesh method is a powerful method to solve eigenequations written in configuration space. It is very easy to implement and very accurate. Using a Gauss quadrature rule, the method requires only the evaluation of the potential at…
We propose a discrete approach for approximating solutions to the prescribed Gaussian curvature problem in two-dimensional manifolds, based on the notion of discrete conformality. Our approach provides an efficient numerical method to…
A novel algorithm for the computation of the quadratic numerical range is presented and exemplified yielding much better results in less time compared to the random vector sampling method. Furthermore, a bound on the probability for the…
We develop an advanced approach for extending Gaussian Differential Privacy (GDP) to general Riemannian manifolds. The concept of GDP stands out as a prominent privacy definition that strongly warrants extension to manifold settings, due to…
Gaussian Processes have become an indispensable part of the spatial statistician's toolbox but are unsuitable for analyzing large dataset because of the significant time and memory needed to fit the associated model exactly. Vecchia…