International School

STOCHASTIC COMPARISONS:
THEORY
AND APPLICATIONS

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.

INVITED TALK:

**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

HOW
TO REACH THE SCHOOL LOCATION

For further information
write to
stochasticorders@calvino.polito.it

**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”