Symmetric and asymmetric theory of relative concentration and applications

Relative concentration theory studies the degree of inequality between two vectors (a(1),....,a(N)) and (alpha (1),....,alpha (N)). It extends concentration theory in the sense that, in the latter theory, one of the above vectors is (1/N,....,1/N) (N coordinates). When studying relative concentration one can consider the vectors (a(1),....,a(N)) and (alpha (1),.....,alpha (N)) as interchangeable (equivalent) or not. In the former case this means that the relative concentration of (a(1),....,a(N)) versus (alpha (1),....,alpha (N)) is the same as the relative concentration of (alpha (1),.....,alpha (N)) versus (a(1),....,a(N)). We deal here with a symmetric theory of relative concentration. In the other case one wants to consider (a(1),....,a(N)) as having a different role as and hence the results can be different when interchanging the vectors. This leads to an asymmetric theory of relative concentration. In this paper we elaborate both models, As they extend concentration theory, both models use the Lorenz order and Lorenz curves. For each theory we present good measures of relative concentration and give applications of each model.