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Covariance selection and multivariate dependence

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  • Bhattacharya, Bhaskar

Abstract

Considering the covariance selection problem of multivariate normal distributions, we show that its Fenchel dual formulation is insightful and allows one to calculate direct estimates under decomposable models. We next generalize the covariance selection to multivariate dependence, which includes MTP2 and trends in longitudinal studies as special cases. The iterative proportional scaling algorithm, used for estimation in covariance selection problems, may not lead to the correct solution under such dependence. Addressing this situation, we present a new algorithm for dependence models and show that it converges correctly using tools from Fenchel duality. We discuss the speed of convergence of the new algorithm. When normality does not hold, we show how to estimate the covariance matrix in an empirical entropy approach. The approaches are compared via simulation and it is shown that the estimators developed here compare favorably with existing ones. The methodology is applied on a real data set involving decreasing CD4+ cell numbers from an AIDS study.

Suggested Citation

  • Bhattacharya, Bhaskar, 2012. "Covariance selection and multivariate dependence," Journal of Multivariate Analysis, Elsevier, vol. 106(C), pages 212-228.
    • Handle: RePEc:eee:jmvana:v:106:y:2012:i:c:p:212-228
    DOI: 10.1016/j.jmva.2011.11.002
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    References listed on IDEAS

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    1. Bhaskar Bhattacharya & Richard Dykstra, 1997. "A Fenchel Duality Aspect of Iterative I-Projection Procedures," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(3), pages 435-446, September.
    2. Rinott, Yosef & Scarsini, Marco, 2006. "Total positivity order and the normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1251-1261, May.
    3. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 499-516, December.
    4. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    5. Sik-Yum Lee, 1980. "Estimation of covariance structure models with parameters subject to functional restraints," Psychometrika, Springer;The Psychometric Society, vol. 45(3), pages 309-324, September.
    6. Nanny Wermuth & Eberhard Scheidt, 1977. "Fitting a Covariance Selection Model to a Matrix," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 26(1), pages 88-92, March.
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