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Bounds for the distribution of the generalized variance. (English) Zbl 0695.62128
Summary: Let \(D_{p,m}\) be the determinant of the sample covariance matrix for \(m+p+1\) observations from a p-variate normal population having identity covariance matrix. We give bounds for the distribution of \(D_{p,m}\) in terms of various chi-squared distribution functions.
Let F(\(\cdot | \nu)\) denote the chi-squared distribution function on \(\nu\) degrees of freedom. We bound \(P\{p(D_{p,m})^{1/p}>t\}\) above by \(1-F(t| p(m+1)+2^{-1}(p-1)(p-2))\) and below by \(1-F(t| p(m+1))\). We give two more bounds involving chi-squared distributions. The proofs use a stochastic analog to the Gauss multiplication theorem.

MSC:
62H10 Multivariate distribution of statistics
33B15 Gamma, beta and polygamma functions
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