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
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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”
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
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