### 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 |
---|---|

Pages (from-to) | 1106-1117 |

Number of pages | 12 |

Journal | Annals of Applied Probability |

Volume | 7 |

Issue number | 4 |

Publication status | Published - Nov 1997 |

Externally published | Yes |

### Fingerprint

### 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

*Annals of Applied Probability*,

*7*(4), 1106-1117.

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

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 7, no. 4, pp. 1106-1117.

}

TY - JOUR

T1 - Inequalities for the probability content of a rotated ellipse and related stochastic domination results

AU - Mathew, Thomas

AU - Nordström, Kenneth

PY - 1997/11

Y1 - 1997/11

N2 - 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.

AB - 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.

KW - Coupling

KW - Majorization

KW - Noncentral chi-square distribution

KW - Poisson mixture representation

KW - Schur-convex function

KW - Stochastic ordering

UR - http://www.scopus.com/inward/record.url?scp=0031260685&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031260685&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031260685

VL - 7

SP - 1106

EP - 1117

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 4

ER -