Perturbation of normal random vectors by nonnormal translations, and an application to HIV latency time distributions
type: Item
instance of: scholarly article
title: Perturbation of normal random vectors by nonnormal translations, and an application to HIV latency time distributions
zbMATH Open document ID: 0844.92021
author: Simeon M. Berman
published in: The Annals of Applied Probability
publication date: 1995-03-30T00:00:00Z
review text: Let \({\mathbf Z}\) be a normal random vector in \(R^k\) and let \textbf{1} be a vector in \(R^k\) with equal components 1. Let \(X\) and \(Y\) be two random variables such that \(X\) and \({\mathbf Z}\) are independent and \(Y\) and \({\mathbf Z}\) are independent. The author studies the bound on the difference between the density functions of \({\mathbf Z} + X \mathbf{1}\) and \({\mathbf Z} + Y\mathbf{1}\). The absolute difference between the two densities is uniformly bounded below by: \[ \bigl[ (2 \pi)^k \text{det} \Sigma \bigr]^{- 1/2} \int^\infty_{- \infty} |E \exp^{i \lambda xX} - E \exp^{i \lambda xY} |\varphi (x) dx, \] where \(\Sigma\) is a nonsingular covariance matrix, \(\varphi (x)\) is the standard normal density and \(\lambda = \sqrt {\mathbf{1}' \Sigma^{-1} \text\textbf{1}}\). This result is applied to a problem related to HIV latency time distributions based on CD4 levels.
Mathematics Subject Classification ID: 92D30
Mathematics Subject Classification ID: 60E99
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 731961
zbMATH Keywords: Gaussian processes
zbMATH Keywords: absolute difference between densities
zbMATH Keywords: normal random vector
zbMATH Keywords: bound
zbMATH Keywords: HIV latency time distributions
zbMATH Keywords: CD4 levels
reviewed by: Yanan Yu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1177004899
OpenAlex ID: W1984011402
DOI: 10.1214/AOAP/1177004899
description: scientific article; zbMATH DE number 731961