INEQUALITIES FOR THE PROBABILITY CONTENT OF A ROTATED ELLIPSE AND RELATED STOCHASTIC DOMINATION RESULTS
成果类型：
Article
署名作者：
Mathew, Thomas; Nordstrom, Kenneth
署名单位：
University System of Maryland; University of Maryland Baltimore County; University of Helsinki
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
发表日期：
1997
页码：
1106-1117
关键词：
摘要：
Let X-i and Y-i follow noncentral chi-square distributions with the same degrees of freedom nu(i) and noncentrality parameters delta(2)(i) and delta(2)(i), 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 Sigma(n)(i=1) lambda(i) X-i is stochastically smaller than Sigma(n)(i=1) lambda(i) Y-i for all nonnegative real numbers lambda(1) >= ... >= 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, ..., X-1 + ... + Xn) is stochastically smaller than (Y-1, Y-1 + Y-2,..., Y-1 + ... + Yn) if and only if Sigma(n)(i=1) lambda(i) X-i is stochastically smaller than Sigma(n)(i=1) lambda(i) Y-i for all nonnegative real numbers lambda(1) >= ... >= lambda(n).