Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Henrion, René; Szantai, Tamas License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-1097


Properties and Calculation of Singular Normal Distributions



The need for calculating and characterizing singular normal distributions arises in a natural way when considering chance constraints of the type P(Az <= b(x) >= p, where A is a rectangular matrix having more rows than columns, b is some function and z is a random vector having some nondegenerate multivariate normal distribution. Such situation is typical, for instance, in stochastic networks, where a comparatively small random vector may induce a possibly large number of linear inequality constraints. Passing to the transformed random variable q:=Az, the constraint can be equivalently rewritten as F(b(x))>= p, where F is the distribution function of q. In contrast to the original random vector z, the transformed vector q has a singular normal distribution. The talk demonstrates how to get back from here to (a sum of) regular normal distributions under a full rank regularity condition. This allows for an efficient calculation of singular normal distributions and provides a numerical method which outperforms competing procedures in moderate dimensions. Computational results for test examples are provided for the sake of comparison. In general, if the mentioned regularity condition is violated, then the singular normal distribution function F may even lack continuity. The talk provides an equivalent criterion for Lipschitz continuity of F and characterizes differentiability and subdifferentiability of F.

BibTeX - Entry

  author =	{Henrion, Ren\'{e} and Szantai, Tamas},
  title =	{{Properties and Calculation of Singular Normal Distributions}},
  booktitle =	{Algorithms for Optimization with Incomplete Information},
  pages =	{1--2},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{5031},
  editor =	{Susanne Albers and Rolf H. M\"{o}hring and Georg Ch. Pflug and R\"{u}diger Schultz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-1097},
  doi =		{10.4230/DagSemProc.05031.24},
  annote =	{Keywords: singular normal distribution , chance constraints , normal probability of polyhedra}

Keywords: singular normal distribution , chance constraints , normal probability of polyhedra
Seminar: 05031 - Algorithms for Optimization with Incomplete Information
Issue date: 2005
Date of publication: 09.06.2005

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