The paper summarizes and extends the theory of generalized ϕ-entropies Hϕ(X) of random variables X obtained as ϕ-informations Iϕ(X;Y) about X maximized over random variables Y. Among the new results is the proof of the fact that these entropies need not be concave functions of distributions pX. An extended class of power entropies Hα(X) is introduced, parametrized by α∈R, where Hα(X) are concave in pX for α≥0 and convex for α<0. It is proved that all power entropies with α≤2 are maximal ϕ-informations Iϕ(X;X) for appropriate ϕ depending on α. Prominent members of this subclass of power entropies are the Shannon entropy H1(X) and the quadratic entropy H2(X). The paper investigates also the tightness of practically important previously established relations between these two entropies and errors e(X) of Bayesian decisions about possible realizations of X. The quadratic entropy is shown to provide estimates which are in average more than 100 \than those based on the Shannon entropy, and this tightness is shown to increase even further when α increases beyond α=2. Finally, the paper studies various measures of statistical diversity and introduces a general measure of anisotony between them. This measure is numerically evaluated for the entropic measures of diversity H1(X) and H2(X).