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

Let Xi and Yi follow noncentral chi-square distributions with the same degrees of freedom .i and noncentrality parameters .2i and .2i, respectively, for i=1,.,n, and let the Xi's be independent and the Yi's independent. A necessary and sufficient condition is obtained under which .ni=1.iXi is stochastically smaller than .ni=1.iYi for all nonnegative real numbers .i....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 (X1,X1+X2,.,X1+.+Xn) is stochastically smaller than (Y1,Y1+Y2,.,Y1+.+Yn) if and only if .ni=1.iXi is stochastically smaller than .ni=1.iYi for all nonnegative real numbers .1....n.