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

Thomas Mathew, Kenneth Nordström

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1106-1117
Number of pages12
JournalAnnals of Applied Probability
Volume7
Issue number4
DOIs
Publication statusPublished - Nov 1997

Keywords

  • Coupling
  • Majorization
  • Noncentral chi-square distribution
  • Poisson mixture representation
  • Schur-convex function
  • Stochastic ordering

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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