Normal approximations for the multivariate inverse Gaussian distribution and asymmetric kernel smoothing on $d$-dimensional half-spaces

Abstract

This paper introduces a novel density estimator supported on $d$-dimensional half-spaces. It stands out as the first asymmetric kernel density estimator for half-spaces in the literature. Using the multivariate inverse Gaussian (MIG) density from Minami (2003) as the kernel and incorporating locally adaptive parameters, the estimator achieves desirable boundary properties. To analyze its mean integrated squared error (MISE) and asymptotic normality, a local limit theorem and probability metric bounds are established between the MIG and the corresponding multivariate Gaussian distribution with the same mean vector and covariance matrix, which may also be of independent interest. Additionally, a new algorithm for generating MIG random vectors is developed, proving to be faster and more accurate than Minami's algorithm based on a Brownian first-hitting location representation. This algorithm is then used to discuss and compare optimal MISE and likelihood cross-validation bandwidths for the estimator in a simulation study under various target distributions. As an application, the MIG asymmetric kernel is used to smooth the posterior distribution of a generalized Pareto model fitted to large electromagnetic storms.

arXiv comment59 pages, 11 figures, 1 table

References in corpus (15)

• Statistical applications of the multivariate skew-normal distribution
• Minimax estimation for mixtures of Wishart distributions
• On multivariate associated kernels for smoothing general density functions
• Asymptotic properties of Dirichlet kernel density estimators
• A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications
• On the boundary properties of Bernstein estimators on the simplex
• Minimax properties of Dirichlet kernel density estimators
• A multivariate normal approximation for the Dirichlet density and some applications
• A refined continuity correction for the negative binomial distribution and asymptotics of the median
• Refined normal approximations for the central and noncentral chi-square distributions and some applications
• Another look at halfspace depth: Flag halfspaces with applications
• Agnostic Learning of Halfspaces with Gradient Descent via Soft Margins
• Robust and Private Learning of Halfspaces
• Refined normal approximations for the Student distribution
• An improvement of Tusnády's inequality in the bulk