International School
June 15-18, 2004
Department of Mathematics - Politecnico di Torino

During the school the following courses will be given:

Moshe Shaked (University of Arizona)
Stochastic orders for lifetime distributions:
What does one want to mean by saying that one random variable X is probabilistically
(or stochastically) smaller than another random variable Y?
Obviously, it should not mean that every realization of X is smaller than every realization
of Y. For example, we all agree that Tucson, Arizona, is a warmer place than Turin.
But occasionally it is cooler in Tucson than in Turin. So the notion of one random variable
being stochastically smaller than another random variable is not a clear and trivial one.
In this course we will see various approaches as to how to define stochastic orders between
two random variables. The notions of the regular stochastic order, the hazard rate order, the
likelihood ratio order, the mean residual order, and some other related orders, will be introduced,
discussed, and compared. Comparisons of multivariate random vectors will also be covered.
Applications in reliability theory and in other areas of applied probability will illustrate the ideas

Alfred Müller (Karlsruhe University)
Variability and dependence orders and their applications in actuarial sciences and finance:
Stochastic orders for the comparison of variability are an important tool with applications in
many fields, especially in economics, finance and insurance in the context of comparing risks.
If there is a portfolio of risks (in mathematical terms a vector of random variables)
then it is also important to investigate the interplay between dependence of the risks and the
risk of the portfolio. In this course we will investigate the most important concepts of
univariate and multivariate variability orders and dependence orders that are useful in this context.

Yosef Rinott (Hebrew University)
Total positivity, orderings and applications:
The notions of Totally Positive matrices and kernels, and the relation to inequalities and
orderings in the univariate and multivariate case. Applications in statistics and other areas will be
discussed. In addition there will be a discussion of orderings in abstract spaces using and
explaining tools from functional analysis. The latter subject is related to coupling of random variables,
an idea which is useful for ordering and inequalities.

Massimo Marinacci and Luigi Montrucchio (Università di Torino)
Ultramodular fuctions

The afternoon of June 18th will be devoted to a small workshop
and round table discussion  with the teachers




For further information write to

Organizing Committee: F. Pellerey, P. Semeraro, B. Trivellato.
Probability and Statistics Group ,
Department of Mathematics, Politecnico di Torino.

Supported by: MIUR-COFIN 2002 “Concepts of Supermodularity in Economic Theory”