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:
| Michael Obermeier | obermeierm@web.de |
| Thomas Augustin | thomas@stat.uni-muenchen.de |
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