Stabilities of generalized entropies

The generalized entropic measure, which is maximized by a given arbitrary distribution under the constraints on normalization of the distribution and the finite ordinary expectation value of a physical random quantity, is considered. To examine if it can be of physical relevance, its experimental robustness is discussed. In particular, Lesche's criterion is analysed, which states that an entropic measure is stable if its change under an arbitrary weak deformation of the distribution (representing fluctuations of experimental data) remains small. It is essential to note the difference between this criterion and thermodynamic stability. A general condition, under which the generalized entropy becomes stable, is derived. Examples known in the literature, including the entropy for the stretched-exponential distribution, the quantum-group entropy and the kappa-entropy are discussed.