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

Thomas Mathew, Kenneth Nordström

Research output: Contribution to journalArticle

4 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
Publication statusPublished - Nov 1997
Externally publishedYes

Fingerprint

Stochastic Domination
Ellipse
Geometric Probability
Non-negative
Noncentral chi-square
Noncentrality Parameter
Chi-square Distribution
Bivariate Distribution
Partial Sums
Extremum
Monotonicity
Gaussian distribution
Complement
Degree of freedom
If and only if
Necessary Conditions
Sufficient Conditions
Domination

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability

Cite this

Inequalities for the probability content of a rotated ellipse and related stochastic domination results. / Mathew, Thomas; Nordström, Kenneth.

In: Annals of Applied Probability, Vol. 7, No. 4, 11.1997, p. 1106-1117.

Research output: Contribution to journalArticle

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