On the Conditional Variance for Scale Mixtures of Normal Distributions

• Published in: Journal of multivariate analysis Vol. 74; no. 2; pp. 163 - 192
• Main Authors: Cambanis, Stamatis, Fotopoulos, Stergios B, He, Lijian
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.08.2000; Elsevier
• Series: Journal of Multivariate Analysis
• Subjects: Distribution theory; Exact sciences and technology; heteroscedasticity; heteroscedasticity, stable random vectors, marginal densities; Linear inference, regression; marginal densities; Mathematics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; stable random vectors; Statistics; Sufficiency and information; heteroscedasticity, stable random vectors, marginal densities; Bessel function; Conditional distribution; Stability; Necessary and sufficient condition; Gaussian distribution; Random vector; Mixture; Heteroscedasticity; Conditional variance; Moment; Marginal distribution; Density function; Zolotarev method
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1006/jmva.1999.1888

For a scale mixture of normal vector, X=A1/2G, where X, G∈Rn and A is a positive variable, independent of the normal vector G, we obtain that the conditional variance covariance, Cov(X2∣X1), is always finite a.s. for m⩾2, where X1∈Rn and m<n, and remains a.s. finite even for m=1, if and only if the square root moment of the scale factor is finite. It is shown that the variance is not degenerate as in the Gaussian case, but depends upon a function SA, m(·) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also given.