## Abstract

Let X_{i} and Y_{i} follow noncentral chi-square distributions with the same degrees of freedom ν_{i} and noncentrality parameters δ^{2}_{i} and μ^{2}_{i}, respectively, for i = 1, . . ., n, and let the X_{i}'s be independent and the Y_{i}'s independent. A necessary and sufficient condition is obtained under which ∑^{n}_{i=1} λ_{i}X_{i} is stochastically smaller than ∑^{n}_{i=1} λ_{i}Y_{i} for all nonnegative real numbers λ_{1 ≥ ⋯ ≥} λ_{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 (X_{1}, X_{1} + X_{2}, . . ., X_{1} +⋯+ X_{n}) is stochastically smaller than (Y_{1}, Y_{1} + Y_{2}, . . ., Y_{1} +⋯+ Y_{n}) if and only if ∑^{n}_{i=1} λ_{i}X_{i} is stochastically smaller than ∑^{n}_{i=1} λ_{i}Y_{i} for all nonnegative real numbers λ_{1 ≥ ⋯ ≥} λ_{n}.

Original language | English |
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Pages (from-to) | 1106-1117 |

Number of pages | 12 |

Journal | Annals of Applied Probability |

Volume | 7 |

Issue number | 4 |

DOIs | |

Publication status | Published - 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