Optimal second-order product probability bounds. (English) Zbl 0782.62057

Block, Henry W.; Costigan, Timothy M.; Sampson, Allan R.

J. Appl. Probab. 30, No. 3, 675-691 (1993).

Summary: Let \(P(c)= P(X_ 1\leq c_ 1,\dots,X_ p\leq c_ p)\) for a random vector \((X_ 1,\dots,X_ p)\). Bounds are considered of the form \[ \beta^ T_ 2(c)= \prod_{(i,j)\in T} P(X_ i\leq c_ i,\;X_ j\leq c_ j)\Bigl/\prod^ p_{i=1} P(X_ i\leq c_ i)^{d_ i-1}, \] where \(T\) is a spanning tree corresponding to the bivariate probability structure and \(d_ i\) is the degree of the vertex \(i\) in \(T\). An optimized version of this inequality is obtained. The main result is that \(\beta^ T_ 2(c)\) always dominates certain second-order Bonferroni bounds. Conditions on the covariance matrix of a \(N(0,\Sigma)\) distribution are given so that this bound applies, and various applications are given.

MSC:

62H05	Characterization and structure theory for multivariate probability distributions; copulas
60E15	Inequalities; stochastic orderings
05C90	Applications of graph theory

Keywords:

product bounds; partial correlation; positive dependence; change point; simultaneous inference; autoregressive correlations; multiple comparisons to a control; group sequential analysis; lower bounds; spanning tree; bivariate probability structure; second-order Bonferroni bounds; covariance matrix