| Title:
|
Asymptotically normal confidence intervals for a determinant in a generalized multivariate Gauss-Markoff model (English) |
| Author:
|
Oktaba, Wiktor |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
40 |
| Issue:
|
1 |
| Year:
|
1995 |
| Pages:
|
55-59 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
By using three theorems (Oktaba and Kieloch [3]) and Theorem 2.2 (Srivastava and Khatri [4]) three results are given in formulas (2.1), (2.8) and (2.11). They present asymptotically normal confidence intervals for the determinant $|\sigma ^2\sum |$ in the MGM model $(U,XB, \sigma ^2\sum \otimes V)$, $ \sum >0$, scalar $\sigma ^2 > 0$, with a matrix $V \ge 0$. A known $n\times p$ random matrix $U$ has the expected value $E(U) = XB$, where the $n\times d$ matrix $X$ is a known matrix of an experimental design, $B$ is an unknown $d\times p$ matrix of parameters and $\sigma ^2\sum \otimes V$ is the covariance matrix of $U,\, \otimes $ being the symbol of the Kronecker product of matrices. A particular case of Srivastava and Khatri’s [4] theorem 2.2 was published by Anderson [1], p. 173, Th. 7.5.4, when $V=I$, $ \sigma ^2 = 1$, $ X=\text{1}$ and $B = \mu ^{\prime } = [\mu _1, \dots , \mu _p]$ is a row vector. (English) |
| Keyword:
|
generalized multivariate Gauss-Markoff model |
| Keyword:
|
singular covariance matrix |
| Keyword:
|
determinant |
| Keyword:
|
asymptotically normal confidence interval |
| Keyword:
|
product of independent chi-squares |
| Keyword:
|
multivariate central limit theorem |
| Keyword:
|
Wishart distribution |
| Keyword:
|
matrix of product sums for error |
| Keyword:
|
hypothesis and “total” |
| MSC:
|
62E20 |
| MSC:
|
62F25 |
| MSC:
|
62H10 |
| MSC:
|
62J99 |
| idZBL:
|
Zbl 0818.62017 |
| idMR:
|
MR1305649 |
| DOI:
|
10.21136/AM.1995.134278 |
| . |
| Date available:
|
2009-09-22T17:46:31Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134278 |
| . |
| Reference:
|
[1] T.W. Anderson: Introduction to Multivariate Statistical Analysis.J. Wiley, New York, 1958. Zbl 0083.14601, MR 0091588 |
| Reference:
|
[2] W. Oktaba: Densities of determinant ratios, their moments and some simultaneous confidence intervals in the multivariate Gauss-Markoff model.Appl. Math. 40 (1995), 47–54. Zbl 0818.62055, MR 1305648 |
| Reference:
|
[3] W. Oktaba, A. Kieloch: Wishart distributions in the multivariate Gauss-Markoff model with singular covariance matrix.Appl. Math. 38 (1993), 61–66. MR 1202080 |
| Reference:
|
[4] M.S. Srivastava, C.G. Khatri: An Introduction to Multivariate Statistics.North Holland, New York, 1979. MR 0544670 |
| . |
