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On computing the dispersion function


A two-dimensional analogue of the well-known bisection method for root finding is presented in order to solve the following problem, related to the dispersion function of a set of random variables: given distribution functionsF 1,...,F n and a probabilityp, find an interval [a,b] of minimum width such thatF i(b)−F i(a )⩾p, fori=1,...,n.

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  1. 1.

    Doeblin, W., andLévy, P.,Sur les Sommes de Variables Aléatoires Indépendant à Dispersions Bornées Inférieurement, Comptes Rendus de l'Academie des Sciences, Paris, Vol. 202, pp. 2027–2029, 1936.

  2. 2.

    Kanter, M.,Probability Inequalities for Convex Sets and Multidimensional Concentration Functions, Journal of Multivariate Analysis, Vol. 6, pp. 222–236, 1976.

  3. 3.

    Kesten, H.,A Sharper Form of the Doeblin-Lévy-Kolomogorov-Rogozin Inequality for Concentration Functions, Mathematica Scandinavica, Vol. 25, pp. 133–144, 1969.

  4. 4.

    Kolmogorov, A. N.,Sur les Propriétés des Fonctions de Concentrations de M. P. Lévy, Annales de l'Institut Henri Poincaré, Vol. 16, pp. 27–34, 1958.

  5. 5.

    Lévy, P.,Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris, France, 1937.

  6. 6.

    Piyavskii, S. A.,An Algorithm for Finding the Absolute Extremum of a Function, USSR Computational Mathematics and Mathematical Physics, Vol. 12, pp. 57–67, 1972.

  7. 7.

    Shubert, B. O.,A Sequential Method Seeking the Global Maximum of a Function, SIAM Journal on Numerical Analysis, Vol. 9, pp. 379–388, 1972.

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The author wishes to thank Dr. I. D. Coope, for helpful advice offered during the preparation of this paper, and the referee, whose comments contributed to a clearer presentation.

Communicated by F. Zirilli

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Wood, G.R. On computing the dispersion function. J Optim Theory Appl 58, 331–350 (1988). https://doi.org/10.1007/BF00939689

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Key Words

  • Distribution function
  • dispersion function
  • concentration function
  • bisection
  • interval minimization