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Summary:
Generalizing the result by Y. L. Tong, a chain of inequalities for probabilities in some types of multivariate distributions is proved. These inequalities embrace a large number of interesting special cases. Nine illustrations are given: cases of multivariate equicorrelated normal, $t,\chi^2$, Poisson, exponential distributions, normal and rank statistics for comparing many treatments with one control, order statistics used in estimating quantiles, and characteristic roots of covariance matrices in certain multiple sampling.
References:
[1] O. J. Dunn: Estimation of the medians for dependent variables. Ann. Math. Statist. 30 (1959), 192-197. DOI 10.1214/aoms/1177706374 | MR 0103563 | Zbl 0090.36404
[2] C. W. Dunnett: A multiple comparison procedure for comparing several treatments with a control. J. Amer. Statist. Assoc. 50 (1955), 1096-1121. DOI 10.1080/01621459.1955.10501294 | Zbl 0066.12603
[3] F. A. Haight: Handbook of the Poisson distribution. J. Wiley & Sons, 1967. MR 0208713 | Zbl 0152.37706
[4] 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
[5] D. R. Jensen: An inequality for a class of bivariate chi-square distributions. J. Amer. Statist. Assoc. 64 (1969), 333-336. DOI 10.1080/01621459.1969.10500978 | MR 0240896 | Zbl 0175.17306
[6] D. R. Jensen M. Q. Jones: Simultaneous confidence intervals for variances. J. Amer. Statist. Assoc. 64 (1969), 324-332. DOI 10.1080/01621459.1969.10500977
[7] 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
[8] 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
[9] R. G. Miller, Jr.: Simultaneous statistical inference. McGraw-Hill Book Company, 1966. MR 0215441 | Zbl 0192.25702
[10] Z. Šidák: Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62 (1967), 626-633. MR 0216666
[11] 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
[12] D. Slepian: The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962), 463-501. DOI 10.1002/j.1538-7305.1962.tb02419.x | MR 0133183
[13] R. G. D. Steel: A multiple comparison rank sum test: treatments versus control. Biometrics 15 (1959), 560-572. DOI 10.2307/2527654 | MR 0108869 | Zbl 0097.13404
[14] R. G. D. Steel: A multiple comparison sign test: treatments versus control. J. Amer. Statist. Assoc. 54 (1959), 767-775. DOI 10.2307/2282500 | MR 0109396 | Zbl 0090.36003
[15] Y. L. Tong: Some probability inequalities of multivariate normal and multivariate t. J. Amer. Statist. Assoc. 65 (1970), 1243-1247. DOI 10.1080/01621459.1970.10481159 | Zbl 0225.62067
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