Complementary means with respect to a nonsymmetric invariant mean

It is known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean $$N\,$$ N such that K is invariant with respect to the mean-type mapping $$\left( M,N\right) ,$$ M , N , which means that $$K\circ \left( M,N\right) =K$$ K ∘ M , N = K and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations.

Adaptive Inference for the Bivariate Mean Function in Functional Data

Inference methods are proposed for the bivariate mean function of a continuous stochastic process with a two-dimensional domain. Nonparametric bivariate estimation is facilitated by thresholded projection estimators. Estimators adapt to the sparsity of the bivariate function. Oracle inequality results are developed to describe the adaptive inference methods. The construction of nonparametric bivariate confidence bands is presented. Implementation results show the applicability of the methods in practice.

AGGREGATION OF PREFERENCES IN CRISP AND FUZZY SETTINGS: FUNCTIONAL EQUATIONS LEADING TO POSSIBILITY RESULTS

We analyze various models introduced in social choice to aggregate individual preferences. We show that on the basis of most of these models there is a system of functional equations such that, in many cases, the origin of impossibility results in a social choice model is the non-existence of a solution for the corresponding system. Among the functional equations considered, we pay a particular attention to general means and associativity, proving that the existence of an associative bivariate mean is equivalent to the existence of a semilatticial partial order. This key result allows us to explain how the knowledge of associative bivariate means can be used to solve social choice paradoxes. In our analysis we deal both with crisp and fuzzy settings.