Bounds for the Distribution of the Generalized Variance

Abstract

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 $F(\cdot \mid \nu)$ denote the chi-squared distribution function on $\nu$ degrees of freedom. We bound $P\{p(D_{p,m})^{1/p} > t\}$ above by $1 - F(t \mid p(m + 1) + \frac{1}{2}(p - 1)(p - 2))$ and below by $1 - F(t \mid p(m + 1))$. We give two more bounds involving chi-squared distributions. The proofs use a stochastic analog to the Gauss multiplication theorem.