On the conditional variance for scale mixtures of normal distributions
type: Item
instance of: scholarly article
title: On the conditional variance for scale mixtures of normal distributions
zbMATH Open document ID: 0962.62010
author: Stamatis Cambanis
author: Lijian He
author: Stergios Fotopoulos
published in: Journal of Multivariate Analysis
publication date: 2001-06-18T00:00:00Z
review text: Suppose \({\mathbb X}\) is a scale mixture of normal vectors, that is \({\mathbb X}=A^{1/2}{\mathbb G}\), where \({\mathbb G\in R^n}\) is a Gaussian random vector, \(n>1\), and \(A\) is a positive variable independent of \({\mathbb G}\). Write \({\mathbb X}=({\mathbb X_1,\mathbb X_2})\) where \({\mathbb X_1\in R^m}\) and \(1\leq m<n\). The authors establish that the conditional second moment of \({\mathbb X_2}\) given \({\mathbb X_1}\) is always finite a.s. for \(m\geq 2\), and it is a.s. finite for \(m=1\) if and only if \(E[A^{1/2}]<\infty\). An expression is derived for the conditional covariance Cov\(({\mathbb X_2 |\mathbb X_1})\) in terms of the distribution of \({\mathbb X}\) and the Laplace transform of the distribution of \(A\). Applications to uniform and stable scales of normal distributions are also given.
Mathematics Subject Classification ID: 62E15
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 60E10
zbMATH DE Number: 1533087
zbMATH Keywords: heteroscedasticity
zbMATH Keywords: stable random vectors
zbMATH Keywords: marginal densities
reviewed by: Alexander Ulanovskii
MaRDI profile type: Publication
full work available at URL: jmva.1999.1888
OpenAlex ID: W2077593638
cites work: Q3331506
cites work: Q4768455
cites work: Q4726487
cites work: Q4884608
cites work: On the theory of elliptically contoured distributions
cites work: Multiple regression on stable vectors
cites work: Elliptically contoured measures on infinite-dimensional Banach spaces
cites work: Q3671491
cites work: Q3040312
cites work: On the linearity of regression
cites work: Q5788558
cites work: Mixtures of distributions, moment inequalities and measures of exponentiality and normality
cites work: Q5628978
cites work: Infinite Divisibility and Variance Mixtures of the Normal Distribution
cites work: Q3209924
cites work: Existence of joint moments of stable random variables
cites work: Metric spaces and completely monontone functions
cites work: Q3763309
cites work: Q3784911
cites work: Q3359479
cites work: Q3326633
DOI: 10.1006/JMVA.1999.1888
description: scientific article; zbMATH DE number 1533087