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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”
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”
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.
References:
[1] T.W. Anderson: Introduction to Multivariate Statistical Analysis. J. Wiley, New York, 1958. MR 0091588 | Zbl 0083.14601
[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 1305648 | Zbl 0818.62055
[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
[4] M.S. Srivastava, C.G. Khatri: An Introduction to Multivariate Statistics. North Holland, New York, 1979. MR 0544670
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