Andrzej Okolewski

On Chebyshev type inequalities for generalized Sugeno integrals

Abstract We give the necessary and sufficient conditions guaranteeing the validity of Chebyshev type inequalities for generalized Sugeno integrals in the case of functions belonging to a much wider class than the comonotone functions. For several choices of operators, we characterize the classes of functions for which the Chebyshev type inequality for the classical Sugeno integral is satisfied.

On the Jensen type inequality for generalized Sugeno integral

We prove necessary and sufficient conditions for the validity of Jensen type inequalities for generalized Sugeno integral. Our proofs make no appeal to the continuity of neither the fuzzy measure nor the operators. For several choices of operators, we characterize the classes of functions for which the corresponding inequalities are satisfied.

Extremal properties of order statistic distributions for dependent samples with partially known multidimensional marginals

Let X=(X1,…,Xn) be an n-tuple of random variables where each Xj has the same known distribution function F and where there is a number k≤n such that for each i∈{1,…,k}, all i-tuples have copulas with the same known diagonal δi. A reliability system with such nonnegative component lifetimes X1,…,Xn is a system with the property that for each i≤k, all of its structurally identical sub-systems of i components have the same known reliability function. We provide a characterization for empirical distributions from the Xj’s, and apply it to derive two-sided bounds (depending on F and δi’s) for arbitrary linear combinations of distribution functions of the associated order statistics as well as to establish necessary and sufficient conditions for uniform sharpness of these bounds. Moreover, for k=2 and some classes of δ2’s, we determine stochastically extremal distributions of single order statistics.