Generalized Entropy Optimized by an Arbitrary Distribution

We construct the generalized entropy optimized by a given arbitrary statistical distribution with a finite linear expectation value of a basic random quantity of interest. This offers, via the maximum entropy principle, a unified basis for a great variety of distributions observed in nature, which can hardly be described by the conventional methods. As a simple example, we explicitly derive the entropy associated with the stretched exponential distribution. To include the distributions with the divergent moments (e.g., the Lévy stable distributions), it is necessary to modify the definition of the expectation value.