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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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