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November 1997 Inequalities for the probability content of a rotated ellipse and related stochastic domination results
Thomas Mathew, Kenneth Nordström
Ann. Appl. Probab. 7(4): 1106-1117 (November 1997). DOI: 10.1214/aoap/1043862426

Abstract

Let Xi and Yi follow noncentral chi-square distributions with the same degrees of freedom νi and noncentrality parameters δi2 and μi2, 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 Σi=1nλiXi is stochastically smaller than Σi=1nλiYi for all nonnegative real numbers λiλn. 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 Σi=1nλiXi is stochastically smaller than Σi=1nλiYi for all nonnegative real numbers λ1λn.

Citation

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Thomas Mathew. Kenneth Nordström. "Inequalities for the probability content of a rotated ellipse and related stochastic domination results." Ann. Appl. Probab. 7 (4) 1106 - 1117, November 1997. https://doi.org/10.1214/aoap/1043862426

Information

Published: November 1997
First available in Project Euclid: 29 January 2003

zbMATH: 0890.60010
MathSciNet: MR1484799
Digital Object Identifier: 10.1214/aoap/1043862426

Subjects:
Primary: 60D05 , 60E15
Secondary: 62H99

Keywords: coupling , majorization , noncentral chi-square distribution , Poisson mixture representation , Schur-convex function , stochastic ordering

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.7 • No. 4 • November 1997
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