Score functions, generalized relative Fisher information and applications

Abstract

Generalizations of the linear score function, a well-known concept in theoretical statistics, are introduced. As the Gaussian density and the classical Fisher information are closely related to the linear score, nonlinear (respectively fractional) score functions allow to identify generalized Gaussian densities (respectively Levy stable laws) as the (unique) probability densities for which the score of a random variable X is proportional to \(-X\). In all cases, it is shown that the variance of the relative to the generalized Gaussian (respectively Levy) score provides an upper bound for \(L^1\)-distance from the generalized Gaussian density (respectively Levy stable laws). Connections with nonlinear and fractional Fokker–Planck type equations are introduced and discussed.