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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.

Abstract Tubes Associated with Perturbed Polyhedra with Applications to Multidimensional Normal Probability Computations