Different Algorithms for Obtaining Upper Bounds to Multivariate Normal Areas Outside of Origin Centered Rectangles Using Joint Marginal Probabilities

摘要

Upper bounds to multivariate normal probability areas outside of n dimensional rectangles centered at the origin are of interest due to their applications in producing conservative simultaneous confidence intervals and hypothesis tests. The current procedure used to compute these upper bounds (Dunn-Sidak method) is based upon making the conservative assumption that the variables are independent. Three new approaches which give tighter (lower) upper bounds for such probability areas have been developed. The first of these (intersection subtraction) is an improved version of the Bonferonni upper bounds. The second of these methods (conditional multiplicative) requires that the multivariate normal distribution have the MTP-2 property. The third method (conservative independent subunit) is a more complicated form of the conservative assumption of independence among variables. These three methods are compared theoretically with the following results: 1) The conditional multiplicative, when it can be applied is better than the other two methods; and 2) The intersection subtraction is better than the conservative independent subunit when n is small, but becomes worse than the conservative independent subunit as n becomes larger. (KR)