Let Xi and Yi follow noncentral chi-square distributions with the same degrees of freedom νi and noncentrality parameters δ2i and μ2i, 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 λiXi is stochastically smaller than ∑ni=1 λiYi for all nonnegative real numbers λ1 ≥ ⋯ ≥ λn. 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 λiXi is stochastically smaller than ∑ni=1 λiYi for all nonnegative real numbers λ1 ≥ ⋯ ≥ λn.
- Noncentral chi-square distribution
- Poisson mixture representation
- Schur-convex function
- Stochastic ordering
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty