Abstract tubes associated with perturbed polyhedra with applications to multidimensional normal probability computations

TL;DR

This paper refines the theory of abstract tubes for perturbed convex polyhedra, providing an efficient method to compute multidimensional normal probabilities for such regions using linear programming and recursive integration.

Contribution

It introduces a new approach to construct abstract tubes for perturbed polyhedra and demonstrates their application in efficient multidimensional normal probability calculations.

Findings

Efficient computation of multidimensional normal probabilities for polyhedral regions.

Implementation of an algorithm using linear programming for abstract tube construction.

Numerical examples illustrating applications to studentized range statistics.

Abstract

Let $K$ be a closed convex polyhedron defined by a finite number of linear inequalities. In this paper we refine the theory of abstract tubes (Naiman and Wynn, 1997) associated with $K$ when $K$ is perturbed. In particular, we focus on the perturbation that is lexicographic and in an outer direction. An algorithm for constructing the abstract tube by means of linear programming and its implementation are discussed. Using the abstract tube for perturbed $K$ combined with the recursive integration technique proposed by Miwa, Hayter and Kuriki (2003), we show that the multidimensional normal probability for a polyhedral region $K$ can be computed efficiently. In addition, abstract tubes and the distribution functions of studentized range statistics are exhibited as numerical examples.