Abstract
Let X 1,...,X d be d (d≥3) dependent random variables with finite variances such that X j ∼F j . Results on the set S d (F 1,...,F d ) of possible correlation matrices with given margins are obtained; this set is relevant for simulating dependent random variables with given marginal distributions and a given correlation matrix. When F 1=...=F d =F, we let S d (F) denote the set of possible correlation matrices. Of interest is the set of F for which S d (F) is the same as the set of all non-negative definite correlation matrices; using a construction with conditional distributions, we show that this property holds only if F is a (location-scale shift of a) margin of a (d−1)-dimensional spherical distribution.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Caudras, C. M. (1992). Probability distributions with given multivariate margins and given dependence structure, Journal of Multivariate Analysis, 42, 51–66.
Davis, P. J., and Rabinowitz, P. (1984). Methods of Numerical Integration, Second edition, Academic Press, Orlando.
Donnelly, T. G. (1973). Algorithm 462: Bivariate normal distribution, Communications of the Association for Computing Machinery, 16, 638.
Emrich, L. J., and Piedmonte, M. R. (1991). A method for generating high-dimensional multivariate binary variates, The American Statistician, 45, 302–304.
Fang, K.-T., Kotz, S., and Ng, K.-W. (1990). Symmetric Multivariate and Related Distributions, Chapman_& Hall, London.
Hoeffding, W. (1940). Maßstabinvariante Korrelationstheorie, Schriftenreihe des Mathematischen Instituts der Universität Berlin, 5, 181–233.
Joe, H. (1997). Multivariate Models and Dependence Concepts, Chapman & Hall, London.
Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale parameter generalization, Sankhyà, Series A, 32, 419–430.
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8, 229–231.
Song, P. X. (1997). Generating dependent random numbers with given correlations and margins from exponential dispersion models, Journal of Statistical Computation and Simulation, 56, 317–335.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Joe, H. (2006). Range of Correlation Matrices for Dependent Random Variables with Given Marginal Distributions. In: Balakrishnan, N., Sarabia, J.M., Castillo, E. (eds) Advances in Distribution Theory, Order Statistics, and Inference. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4487-3_8
Download citation
DOI: https://doi.org/10.1007/0-8176-4487-3_8
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4361-4
Online ISBN: 978-0-8176-4487-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)