Summary
In the present paper we deal with the characterization of some dependence concepts for the multivariate normal distribution. It turns out that normal distributions have some special properties w.r.t. these dependence concepts and, furthermore, that the characterizations are closely connected to some interesting problems on matrices. Some applications to simultaneous confidence bounds are discussed.
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References
Abdel-Hameed, M. and Sampson, A. R. (1978). Positive dependence of the bivariate and trivariate absolute normalt, χ2 andF distributions.Ann. Statist.,6, 1360–1368.
Barlow, R. E. and Proschan, F. (1975).Statistical Theory of Reliability and Life Testing, Holt, New York.
Das Gupta, S., Eaton, M. L., Olkin, I., Perlman, M. D., Savage, L. J. and Sobel, M. (1972). Inequalities on the probability content of convex regions for elliptically contoured distributions,Proc. Sixth Berkeley Symp. Math. Statist. Prob., 2, Univ. of California Press, Berkeley California, 241–265.
Dunn, O. J. (1958). Estimation of the means of dependent variables,Ann. Math. Statist.,29, 1095–1110.
Dykstra, R. L., Hewett, J. E. and Thompson, W. A., Jr. (1973). Events which are almost independent,Ann. Statist.,1, 674–681.
Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications,Ann. Math. Statist.,38, 1466–1474.
Fiedler, M. and Ptak, V. (1962). On matrices with non-positive off-diagonal elements and positive principal minors,Czech. Math. J.,12, 382–400.
Gray, L. J. and Wilson, D. G. (1980). Nonnegative factorization of positive semidefinite nonnegative matrices,Linear Algebra and Appl.,31, 119–127.
Hall, M., Jr. and Newman, M. (1963). Copositive and completely positive quadratic forms,Proc. Camb. Phil. Soc.,59, 329–339.
Jacobsen, D. H. (1974). Factorization of symmetricM-matrices,Linear Algebra and Appl.,9, 275–278.
Jogdeo, K. (1977). Association and probability inequalities,Ann. Statist.,5, 495–504.
Khatri, C. G. (1967). On certain inequalities for normal distributions and their applications to simultaneous confidence bouuds,Ann. Math. Statist.,38, 1853–1867.
Lehmann, E. L. (1966). Some concepts of dependence,Ann. Math. Statist.,37, 1137–1153.
Markham, T. L. (1971). Factorization of completely positive matrices,Proc. Camb. Phil. Soc.,69, 53–58.
Plesken, W., Rüschendorf, L. and Krafft, O. (1980). Nonnegative factorization of nonnegative definite, nonnegative matrices, unpublished manuscript.
Poole, G. and Boullion, T. (1974). A survey onM-matrices,SIAM Rev.,16, 419–427.
Rüschendorf, L. (1980). Weak association of random variables, to appear inJ. Multivariate Anal.
Sarkar, T. K. (1969). Some lower bounds of reliability,Technical Report No. 124, Department of Operations Research and Statistics, Stanford University, California.
Scott, A. (1967). A note on conservative regions for the mean of a multivariate normal,Ann. Math. Statist.,38, 278–280.
Sidak, Z. (1975). A note on C. G. Khatri and A. Scott's papers on multivariate normal distributions,Ann. Inst. Statist. Math.,27, 181–184.
Slepian, D. (1962). The one-sided barrier problem for Gaussian noise,Bell Syst. Tech. J.,41, 463–501.
Srivastava, M. S. and Khatri, C. G. (1979).An Introduction to Multivariate Statistics, North Holland, New York.
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Rüschendorf, L. Characterization of dependence concepts in normal distributions. Ann Inst Stat Math 33, 347–359 (1981). https://doi.org/10.1007/BF02480946
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DOI: https://doi.org/10.1007/BF02480946