Comparison theorems for distribution functions of quadratic forms of Gaussian vectors

Let $Q_1$ and $Q_2$ be nonnegatively definite quadratic forms of centered Gaussian random variables (r.v.'s) satisfying normalization condition $\mathbf{E}Q_1={\mathbf E}Q_2=1$. If the vector of eigenvalues of $Q_1$ majorizes that of $Q_2$, then the distribution function of $Q_1$ is less than the distribution function of $Q_2$ when their arguments exceed 2. Some statistical applications are given.