Product Inequalities Involving the Multivariate Normal Distribution

References

• Anderson , T. W. “The Integral of a Symmetric Unimodal Function Over a Symmetric Convex Set and Some Probability Inequalities,” . Proceedings of the American Mathematical Society . Vol. 6 , pp. 170 – 176 .
   Google Scholar
• Anderson , T. W. 1958 . An Introduction to Multivariate Statistical Analysis , New York : John Wiley & Sons .
   Google Scholar
• Billingsley , Patrick . 1968 . Convergence of Probability Measures , New York : John Wiley & Sons .
   Google Scholar
• Gupta , Das S. , Eaton , M. L. , Olkin , I. , Perlman , M. , Savage , L. J. and Sobel , M. “Inequalities on the Probability Content of Convex Regions for Elliptically Contoured Distributions,” . Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability . Vol. II , pp. 241 – 265 .
   Google Scholar
• Dunn , Olive Jean . 1958 . “Estimation of the Means of Dependent Variables,” . Annals of Mathematical Statistics , 29 : 1095 – 1111 .
   Google Scholar
• Dykstra , R. L. and Hewett , J. E. 1978 . “Positive Dependence of the Roots of a Wishart Matrix,” . Annals of Statistics , 6 : 235 – 238 .
   Web of Science ®Google Scholar
• Dykstra , R. L. , Hewett , J. E. and Thompson , W. A. Jr. 1973 . “Events Which Are Almost Independent,” . Annals of Statistics , 1 : 674 – 681 .
   Web of Science ®Google Scholar
• Esary , J. D. , Proschan , F. and Walkup , D. W. 1967 . “Association of Random Variables, With Applications,” . Annals of Mathematical Statistics , 38 : 1466 – 1474 .
   Google Scholar
• Jensen , D. R. 1969 . “An Inequality for a Class of Bivariate Chi-Square Distributions,” . Journal of the American Statistical Association , 64 : 333 – 336 .
   Web of Science ®Google Scholar
• Jogdeo , Kumar . 1970 . “A Simple Proof of an Inequality for Multivariate Normal Probabilities of Rectangles,” . Annals of Mathematical Statistics , 41 : 1357 – 1359 .
   Google Scholar
• Khatri , C. G. 1967 . “On Certain Inequalities for Normal Distributions and Their Applications to Simultaneous Confidence Bounds,” . Annals of Mathematical Statistics , 38 : 1853 – 1867 .
   Google Scholar
• Khatri , C. G. 1970 . “Further Contributions to Some Inequalities for Normal Distributions and Their Applications to Simultaneous Confidence Bounds,” . Annals of the Institute of Statistical Mathematics , 22 : 451 – 458 .
   Google Scholar
• Kimball , A. W. 1951 . “On Dependent Tests of Significance in an Analysis of Variance,” . Annals of Mathematical Statistics , 22 : 600 – 602 .
   Google Scholar
• Lehmann , E. L. 1966 . “Some Concepts of Dependence,” . Annals of Mathematical Statistics , 37 : 1137 – 1153 .
   Google Scholar
• Marshall , Albert W. and Olkin , Ingram . 1974 . “Majorization in Multivariate Distributions,” . Annals of Statistics , 6 : 1189 – 1200 .
   Google Scholar
• Pitt , Loren D. 1977 . “A Gaussian Correlation Inequality for Symmetric Convex Sets,” . Annals of Probability , 5 : 470 – 474 .
   Web of Science ®Google Scholar
• Schuurmann , F. J. , Krishnaiah , P. R. and Chattopadhyay , A. K. 1975 . “Tables for a Multivariate F distribution,” . Sankyā , 37 : 308 – 331 . Ser. B
   Google Scholar
• Scott , Alastair . 1967 . “A Note on Conservative Confidence Regions for the Mean of a Multivariate Normal,” . Annals of Mathematical Statistics , 38 : 278 – 280 .
   Google Scholar
• Shaked , Moshe . 1977 . “A Concept of Positive Dependence for Exchangeable Random Variables,” . Annals of Statistics , 5 : 505 – 515 .
   Web of Science ®Google Scholar
• Sidak , Z. 1967 . “Rectangular Confidence Regions for the Means of Multivariate Normal Distributions,” . Journal of the American Statistical Association , 68 : 734 – 739 .
   Google Scholar
• Sidak , Z. 1968 . “On Multivariate Normal Probabilities of Rectangles: Their Dependence on Correlations,” . Annals of Mathematical Statistics , 39 : 1425 – 1434 .
   Google Scholar
• Sidak , Z. 1971 . “On Probabilities of Rectangles in Multivariate Student Distributions: Their Dependence on Correlations,” . Annals of Mathematical Statistics , 42 : 169 – 175 .
   Google Scholar
• Sidak , Z. 1973 . “A Chain of Inequalities for Some Types of Multivariate Distributions, With Nine Special Cases,” . Aplikace Matematiky , 18 : 110 – 118 .
   Google Scholar
• Sidak , Z. 1975 . “A Note on C.G. Khatri's and A. Scott's Papers on Multivariate Normal Distributions,” . Annals of the Institute of Statistical Mathematics , 27 : 181 – 184 .
   Web of Science ®Google Scholar
• Siotani , M. 1959 . “The Extreme Value of the Generalized Distances of the Individual Points in the Multivariate Normal Sample,” . Annals of the Institute of Statistical Mathematics , 10 : 183 – 203 .
   Web of Science ®Google Scholar
• Slepian , D. 1962 . “The One-Sided Barrier Problem for Gaussian Noise,” . Bell System Technical Journal , 41 : 463 – 501 .
   Google Scholar
• Tong , Y. L. 1970 . “Some Probability Inequalities of Multivariate Normal and Multivariate t,” . Journal of the American Statistical Association , 65 : 1243 – 1247 .
   Google Scholar