Asymptotically normal confidence intervals for a determinant in a generalized multivariate Gauss-Markoff model

Oktaba, Wiktor

Applications of Mathematics, Tome 40 (1995) no. 1, pp. 55-59.

Résumé

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.

Détail

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”