| Title:
|
On probabilities in certain multivariate distributions: their dependence on correlations (English) |
| Author:
|
Šidák, Zbyněk |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
18 |
| Issue:
|
2 |
| Year:
|
1973 |
| Pages:
|
128-135 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| 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. () |
| MSC:
|
60E05 |
| MSC:
|
62H05 |
| MSC:
|
62H10 |
| idZBL:
|
Zbl 0261.62041 |
| idMR:
|
MR0314197 |
| DOI:
|
10.21136/AM.1973.103459 |
| . |
| Date available:
|
2008-05-20T17:55:43Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103459 |
| . |
| Reference:
|
[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. Zbl 0066.37402, MR 0069229, 10.1090/S0002-9939-1955-0069229-1 |
| Reference:
|
[2] F. A. Haight: Handbook of the Poisson distribution.J. Wiley & Sons, 1967. Zbl 0152.37706, MR 0208713 |
| Reference:
|
[3] P. Holgate: Estimation for the bivariate Poisson distribution.Biometrika 51 (1964), 241 - 245. Zbl 0133.11802, MR 0172374, 10.1093/biomet/51.1-2.241 |
| Reference:
|
[4] C. G. Khatri: On certain inequalities for normal distributions and their applications to simultaneous confidence bounds.Ann. Math. Statist. 38 (1967), 1853 - 1867. Zbl 0155.27103, MR 0220392, 10.1214/aoms/1177698618 |
| Reference:
|
[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. Zbl 0294.62013, MR 0283913, 10.1007/BF02506363 |
| Reference:
|
[6] A. W. Marshall I. Olkin: A multivariate exponential distribution.J. Amer. Statist. Assoc. 62 (1967), 30-44. MR 0215400, 10.1080/01621459.1967.10482885 |
| Reference:
|
[7] Z. Šidák: Unequal numbers of observations in comparing several treatments with one control.(In Czech.) Apl. Mat. 7 (1962), 292-314. |
| Reference:
|
[8] Z. Šidák: On probabilities of rectangles in multivariate Student distributions: their dependence on correlations.Ann. Math. Statist. 42 (1971), 169-175. MR 0278354, 10.1214/aoms/1177693504 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| . |
