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- W2089121903 abstract "Let $X_i$ and $Y_i$ follow noncentral chi-square distributions with the same degrees of freedom $nu_i$ and noncentrality parameters $delta_i^2$ and $mu_i^2$, respectively, for $i = 1, dots, n$, and let the $X_i$'s be independent and the $Y_i$'s independent. A necessary and sufficient condition is obtained under which $Sigma_{i = 1}^n lambda_i X_i$ is stochastically smaller than $Sigma_{i = 1}^n lambda_i Y_i$ for all nonnegative real numbers $lambda_i geq dots geq lambda_n$. 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 $(X_1, X_1 + X_2, dots, X_1 + dots + X_n)$ is stochastically smaller than $(Y_1, Y_1 + Y_2, dots, Y_1 + dots + Y_n)$ if and only if $Sigma_{i=1}^n lambda_i X_i$ is stochastically smaller than $Sigma_{i=1}^n lambda_i Y_i$ for all nonnegative real numbers $lambda_1 geq dots geq lambda_n$." @default.
- W2089121903 created "2016-06-24" @default.
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- W2089121903 date "1997-11-01" @default.
- W2089121903 modified "2023-09-26" @default.
- W2089121903 title "Inequalities for the probability content of a rotated ellipse and related stochastic domination results" @default.
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- W2089121903 doi "https://doi.org/10.1214/aoap/1043862426" @default.
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