Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-01T03:13:53.099Z Has data issue: false hasContentIssue false
  • English
  • Français

Entropy inequalities for classes of probability distributions II. The multivariate case

Published online by Cambridge University Press:  01 July 2016

Samuel Karlin  and
Yosef Rinott
Samuel Karlin*
Affiliation:
Stanford University
Yosef Rinott*
Affiliation:
Stanford University
*
∗Postal adress: Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A.
∗Postal adress: Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we continue our investigation of entropy comparisons with emphasis on multivariate distributions. For multiparameter cases of the multinomial and negative multinomial distributions we consider various higher-order forms of multivariate convexity. For the multinormal, Wishart, and t-distributions we define a partial ordering on the set of covariance matrices and determine monotonicity of the entropy functional. We further indicate some entropy inequalities for different sampling schemes. Because of the complex nature of multivariate partial ordering relations several problems remain open.

Keywords

ENTROPY FUNCTIONALMULTIVARIATE DISTRIBUTIONSMULTIVARIATE SCHUR CONVEXITY
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 325 - 351
Copyright
Copyright © Applied Probability Trust 1981 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Yosef Rinott is also a member of the Department of Statistics, The Hebrew University, Jerusalem, Israel.

Supported in part by NIH Grant 2R01 GM10452–16 and NSF Grant MCS76–80624-A02.

References

Cheng, K. W. (1977) Majorization: Its extensions and preservation theorems Tech. Report No. 121, Dept. of Statist., Stanford University.Google Scholar
Davis, P. J. (1964) Gamma functions and related functions. In Handbook of Mathematical Functions , ed. Abramowitz, M. and Stegun, I. A. N.B.S. Applied Math. Series 55, 253–293.Google Scholar
Eaton, M. L. and Perlman, M. D. (1977) Reflection groups generalized Schur functions and the geometry of majorization. Ann. Prob. 5, 829–860.CrossRefGoogle Scholar
Hardy, G. H., Littlewood, J. E. and Pólya, G. (1934) Inequalities. Cambridge University Press Google Scholar
Jogdeo, K. (1977) Association and probability inequalities. Ann. Statist. 5, 495–504.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics: Discrete Distributions. Wiley, New York.Google Scholar
Johnson, N. L. and Kotz, S. (1972) Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.Google Scholar
Karlin, S. (1974) Inequalities for symmetric sampling plans I. Ann. Statist. 2, 1065–1094.Google Scholar
Karlin, S. and Rinott, Y. (1981) Entropy inequalities for classes of probability distributions: I. The univariate case. Adv. Appl. Prob, 13, 93–112.CrossRefGoogle Scholar
Karlin, S. and Rinott, Y. (1982) Multivariate monotonicity, generalized convexity and ordering relations, with applications in analysis and statistics. In preparation.Google Scholar
Marcus, M. and Minc, H. (1964) A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Inc., Boston.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities, Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Mudholkar, G. S. (1966) The integral of an invariant unimodal function over an invariant convex set—an inequality and applications. Proc. Amer. Math. Soc. 17, 1327–1333.Google Scholar
Parthasarathy, K. R. and Schmidt, K. (1972) Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory. Springer-Verlag, Berlin.Google Scholar
Rinott, Y. (1973) Multivariate majorization and rearrangement inequalities with some applications to probability and statistics. Israel J. Math. 15, 60–77.Google Scholar