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ISIPTA'07 -
FIFTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Charles University, Faculty of Mathematicsand Physics

Prague, Czech Republic

16-19 July 2007

## ELECTRONIC PROCEEDINGS

## Michael Obermeier, Thomas Augustin

# Lucenos discretization methods and its application in decision making under ambiguity

### Abstract

When extending classical statistical models to imprecise
probabilities, one fundamental difficulty, which
may have hindered some powerful practical applications,
is the following gap: While classical statistical
models are typically based on absolutely continuous
probability distributions, most computational methods
developed for handling imprecise probability models
rely on finite sample spaces. A natural way to close
this gap is discretization of the underlying continuous
probability distribution. This, however, is far from
straightforward, because na¨ıve discretization by mere
rounding may cause a substantial bias; even moments
of very low order would be distorted. The present paper
discusses the application of Luce˜nos’ ([10]) so-tosay
adaptive discretization method in imprecise probability
models. We firstly recall two theorems, showing,
for any fixed natural number r, how to construct
a discrete random variable such that its first, second,
... r-th moment coincides with the corresponding moment
of the underlying continuous distribution. (In
addition, also coincidence of the distribution functions
in a fixed number of points can be enforced.) Then
we illustrate the power of the method by utilizing it
in decision problems under ambiguity.

** Keywords. ** Decision making under ambiguity, discretization, Gaussian quadrature, imprecise probabilities, interval probability, linear programming, Luceno, numerical integration

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** Authors addresses: **

Michael Obermeier

Frankplatz 18,

80939 München

Thomas Augustin

Department of Statistics

University of Munich

Ludwigstr. 33

D-80539 Munich

Germany

** E-mail addresses: **

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