Geodesic
Asymptotically normal confidence intervals for a determinant in a generalized multivariate Gauss-Markoff model
Applications of Mathematics, Tome 40 (1995) no. 1, pp. 55-59

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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.
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.
DOI : 10.21136/AM.1995.134278
Classification : 62E20, 62F25, 62H10, 62J99
Keywords: generalized multivariate Gauss-Markoff model; singular covariance matrix; determinant; asymptotically normal confidence interval; product of independent chi-squares; multivariate central limit theorem; Wishart distribution; matrix of product sums for error; hypothesis and “total” 
Oktaba, Wiktor. Asymptotically normal confidence intervals for a determinant in a generalized multivariate Gauss-Markoff model. Applications of Mathematics, Tome 40 (1995) no. 1, pp. 55-59. doi: 10.21136/AM.1995.134278
@article{10_21136_AM_1995_134278,
     author = {Oktaba, Wiktor},
     title = {Asymptotically normal confidence intervals for a determinant in a generalized multivariate {Gauss-Markoff} model},
     journal = {Applications of Mathematics},
     pages = {55--59},
     year = {1995},
     volume = {40},
     number = {1},
     doi = {10.21136/AM.1995.134278},
     mrnumber = {1305649},
     zbl = {0818.62017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1995.134278/}
}
TY  - JOUR
AU  - Oktaba, Wiktor
TI  - Asymptotically normal confidence intervals for a determinant in a generalized multivariate Gauss-Markoff model
JO  - Applications of Mathematics
PY  - 1995
SP  - 55
EP  - 59
VL  - 40
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/AM.1995.134278/
DO  - 10.21136/AM.1995.134278
LA  - en
ID  - 10_21136_AM_1995_134278
ER  - 
%0 Journal Article
%A Oktaba, Wiktor
%T Asymptotically normal confidence intervals for a determinant in a generalized multivariate Gauss-Markoff model
%J Applications of Mathematics
%D 1995
%P 55-59
%V 40
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1995.134278/
%R 10.21136/AM.1995.134278
%G en
%F 10_21136_AM_1995_134278

[1] T.W. Anderson: Introduction to Multivariate Statistical Analysis. J. Wiley, New York, 1958. | MR | Zbl

[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. | MR | Zbl

[3] W. Oktaba, A. Kieloch: Wishart distributions in the multivariate Gauss-Markoff model with singular covariance matrix. Appl. Math. 38 (1993), 61–66. | MR

[4] M.S. Srivastava, C.G. Khatri: An Introduction to Multivariate Statistics. North Holland, New York, 1979. | MR

Cité par Sources :