13 research outputs found
On the estimation of Pareto fronts from the point of view of copula theory
International audienceGiven a first set of observations from a design of experiments sampled randomly in the design space, the corresponding set of non-dominated points usually does not give a good approximation of the Pareto front. We propose here to study this problem from the point of view of multivariate analysis, introducing a probabilistic framework with the use of copulas. This approach enables the expression of level lines in the objective space, giving an estimation of the position of the Pareto front when the level tends to zero. In particular, when it is possible to use Archimedean copulas, analytical expressions for Pareto front estimators are available. Several case studies illustrate the interest of the approach, which can be used at the beginning of the optimization when sampling randomly in the design space
Comportement extrémal des copules diagonales et de Bertino
The maximal attractors of bivariate diagonal and Bertino copulas are determined under suitable regularity conditions. Some consequences of these facts are drawn, namely bounds on the maximal attractor of a symmetric copula with a given diagonal section, and bounds on Spearman’s rho and Kendall’s tau for an exchangeable extreme-value copula whose upper-tail dependence coefficient is known. Some of these results are then extended to the case of arbitrary bivariate copulas and to multivariate copulas
Comportement extrémal des copules diagonales et de Bertino
The maximal attractors of bivariate diagonal and Bertino copulas are determined under suitable regularity conditions. Some consequences of these facts are drawn, namely bounds on the maximal attractor of a symmetric copula with a given diagonal section, and bounds on Spearman’s rho and Kendall’s tau for an exchangeable extreme-value copula whose upper-tail dependence coefficient is known. Some of these results are then extended to the case of arbitrary bivariate copulas and to multivariate copulas
Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes
As a motivating problem, we aim to study some special aspects of the marginal
distributions of the order statistics for exchangeable and (more generally) for
minimally stable non-negative random variables . In any case,
we assume that are identically distributed, with a common
survival function and their survival copula is denoted by .
The diagonal's and subdiagonals' sections of , along with ,
are possible tools to describe the information needed to recover the laws of
order statistics.
When attention is restricted to the absolutely continuous case, such a joint
distribution can be described in terms of the associated multivariate
conditional hazard rate (m.c.h.r.) functions. We then study the distributions
of the order statistics of also in terms of the system of the
m.c.h.r. functions. We compare and, in a sense, we combine the two different
approaches in order to obtain different detailed formulas and to analyze some
probabilistic aspects for the distributions of interest. This study also leads
us to compare the two cases of exchangeable and minimally stable variables both
in terms of copulas and of m.c.h.r. functions. The paper concludes with the
analysis of two remarkable special cases of stochastic dependence, namely
Archimedean copulas and load sharing models. This analysis will allow us to
provide some illustrative examples, and some discussion about peculiar aspects
of our results