  Density-based directed distances -- particularly known as divergences --
between probability distributions are widely used in statistics as well as in
the adjacent research fields of information theory, artificial intelligence and
machine learning. Prominent examples are the Kullback-Leibler information
distance (relative entropy) which e.g. is closely connected to the omnipresent
maximum likelihood estimation method, and Pearson's chisquare-distance which
e.g. is used for the celebrated chisquare goodness-of-fit test. Another line of
statistical inference is built upon distribution-function-based divergences
such as e.g. the prominent (weighted versions of) Cramer-von Mises test
statistics respectively Anderson-Darling test statistics which are frequently
applied for goodness-of-fit investigations; some more recent methods deal with
(other kinds of) cumulative paired divergences and closely related concepts. In
this paper, we provide a general framework which covers in particular both the
above-mentioned density-based and distribution-function-based divergence
approaches; the dissimilarity of quantiles respectively of other statistical
functionals will be included as well. From this framework, we structurally
extract numerous classical and also state-of-the-art (including new)
procedures. Furthermore, we deduce new concepts of dependence between random
variables, as alternatives to the celebrated mutual information. Some
variational representations are discussed, too.
