# Article

Full entry | PDF   (1.0 MB)
Summary:
First, under a multivariate normal distribution with all correlations of the form $Q_{ij}=b_ib_j$ (where $-1\leq b_i, b_j\leq 1$), the probabilities of certain convex symmetric regions are shown to be, roughly speaking, non-decreasing functions of $\left|Q_{ij}\right|$. Second, under an equicorrelated normal distribution, the probabilieties of certain regions (which need be neither convex nor symmetric) are shown to be non-decreasing functions of the correlations. Third, some inequalities for special cases of multivariate exponential and Poisson distributions are given.
References:
[1] T. W. Anderson: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 (1955), 170-176. DOI 10.1090/S0002-9939-1955-0069229-1 | MR 0069229 | Zbl 0066.37402
[2] F. A. Haight: Handbook of the Poisson distribution. J. Wiley & Sons, 1967. MR 0208713 | Zbl 0152.37706
[3] P. Holgate: Estimation for the bivariate Poisson distribution. Biometrika 51 (1964), 241 - 245. DOI 10.1093/biomet/51.1-2.241 | MR 0172374 | Zbl 0133.11802
[4] C. G. Khatri: On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist. 38 (1967), 1853 - 1867. DOI 10.1214/aoms/1177698618 | MR 0220392 | Zbl 0155.27103
[5] C. G. Khatri: Further contributions to some inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Inst. Statist. Math. 22 (1970), 451 - 458. DOI 10.1007/BF02506363 | MR 0283913 | Zbl 0294.62013
[6] A. W. Marshall I. Olkin: A multivariate exponential distribution. J. Amer. Statist. Assoc. 62 (1967), 30-44. DOI 10.1080/01621459.1967.10482885 | MR 0215400
[7] Z. Šidák: Unequal numbers of observations in comparing several treatments with one control. (In Czech.) Apl. Mat. 7 (1962), 292-314.
[8] Z. Šidák: On probabilities of rectangles in multivariate Student distributions: their dependence on correlations. Ann. Math. Statist. 42 (1971), 169-175. DOI 10.1214/aoms/1177693504 | MR 0278354
[9] Z. Šidák: A note on C. G. Khatri's and A. Scott's papers on multivariate normal distributions. Submitted to Ann. Inst. Statist. Math. Zbl 0368.62029
[10] Z. Šidák: A chain of inequalities for some types of multivariate distributions, with nine special cases. Apl. Mat. 18 (1973), 110-118. MR 0315842

Partner of