Abstract
Inequalities for the probability content \( P\left[ { \cap _{j\, = \,1}^n\left\{ {\,{a_{1j\,}}\, \leqslant \,{X_j}\, \leqslant \,{a_{2j}}} \right\}} \right] \) are obtained, via concepts of multivariate majorization (which involves the diversity of elements of the 2xn matrix A = (a ij)). A special case of the general result is that \( P\left[ { \cap _{j = 1}^n\left\{ {{a_{1j}} \leqslant {X_j} \leqslant {a_{2j}}} \right\}} \right] \leqslant P\left[ { \cap _{j = 1}^n\left\{ {{{\bar a}_1} \leqslant {X_j} \leqslant {{\bar a}_2}} \right\}} \right] \) for \( {\bar a_{i\,}} = \,\frac{1}{n}\,\sum\nolimits_{j = 1}^n {{a_{ij}}\,\left( {i\, = \,1,\,2} \right).} \). The main theorems apply in most important cases, including the exchangeable normal, t, chi-square and gamma, F, beta, and Dirichlet distributions. The proofs of the inequalities involve a convex combination of an n-dimensional rectangle and its permutation sets.
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
Karlin, S. and Rinott, Y. (1981). Entropy inequalities for classes of probability distributions II. The multivariate case. Adv. Appl Probability 13, 325–351.
Karlin, S. and Rinott, Y. (1983). Comparison of measures, multivariate majorization and applications to statistics. In Studies in Econometrics, Time Series and Multivariate Analysis, S. Karlin, T. Amemiya and L. Goodman, eds., Academic Press, New York, 465–489.
Marshall, A.W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
Olkin, I. (1972). Monotonicity properties of Dirichlet integrals with applications to the multinomial distributions and the analysis of variance. Biometrika 59, 303–307.
Prékopa, A. (1971). Logarithmic concave measures with applications. Acta Sci. Math. 32, 301–316.
Rinott, Y. (1973). Multivariate majorization and rearrangement inequalities with some applications to probability and statistics. Israel J. Math. 15, 60–67.
Tong, Y.L. (1980). Probability Inequalities in Multivariate Distributions. Academic Press, New York.
Tong, Y.L. (1982). Rectangular and elliptical probability inequalities for Schur-concave random variables. Ann. Statists. 10, 637–642.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Tong, Y.L. (1989). Probability Inequalities for n-Dimensional Rectangles via Multivariate Majorization. In: Gleser, L.J., Perlman, M.D., Press, S.J., Sampson, A.R. (eds) Contributions to Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3678-8_11
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3678-8_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8200-6
Online ISBN: 978-1-4612-3678-8
eBook Packages: Springer Book Archive