Multivariate Sobel-Uppuluri-Galambos-type bounds (Q2722159)

The upper and the lower bounds in the recent multivariate generalization [\textit{J. Galambos} and \textit{Y. Xu}, in: Statistical theory and applications, 29-36 (1996; Zbl 0853.60016)] of the univariate Sobel-Uppuluri-Galambos inequalities [\textit{M. Sobel} and \textit{V. R. R. Uppuluri}, Ann. Math. Stat. 43, 1549-1558 (1972; Zbl 0254.60013)] are shown to be weighted averages of the individual multivariate bounds and hence can be sharpened by optimizing these individual bounds. Examples are used to illustrate the improvement as well as to illustrate the difference between bounds of this type and the type of multivariate bounds appearing in [\textit{T. Chen} and \textit{E. Seneta}, J. Appl. Probab. 33, No. 3, 729-740 (1996; Zbl 0865.60013) and \textit{J. Galambos} and \textit{M.-Y. Lee}, in: Studies in applied probability, 63-69 (1994; Zbl 0803.60020)]. In this paper, only bounds for the probability of the intersection of exactly \(k_{i}\) events occurring from the \(i\)th event group of \(n_{i}\) events, \(i=1,\dots,d\), are considered.