Inequalities for the probability content of a rotated ellipse and related stochastic domination results

Abstract

Let $X_i$ and $Y_i$ follow noncentral chi-square distributions with the same degrees of freedom $\nu_i$ and noncentrality parameters $\delta_i^2$ and $\mu_i^2$, respectively, for $i = 1, \dots, n$, and let the $X_i$'s be independent and the $Y_i$'s independent. A necessary and sufficient condition is obtained under which $\Sigma_{i = 1}^n \lambda_i X_i$ is stochastically smaller than $\Sigma_{i = 1}^n \lambda_i Y_i$ for all nonnegative real numbers $\lambda_i \geq \dots \geq \lambda_n$. Reformulating this as a result in geometric probability, solutions are obtained, in particular, to the problems of monotonicity and location of extrema of the probability content of a rotated ellipse under the standard bivariate Gaussian distribution. This complements results obtained by Hall, Kanter and Perlman who considered the behavior of the probability content of a square under rotation. More generally, it is shown that the vector of partial sums $(X_1, X_1 + X_2, \dots, X_1 + \dots + X_n)$ is stochastically smaller than $(Y_1, Y_1 + Y_2, \dots, Y_1 + \dots + Y_n)$ if and only if $\Sigma_{i=1}^n \lambda_i X_i$ is stochastically smaller than $\Sigma_{i=1}^n \lambda_i Y_i$ for all nonnegative real numbers $\lambda_1 \geq \dots \geq \lambda_n$.