Kybernetika 39 no. 1, 29-42, 2003

Core functions and core divergences of regular distributions

Zdeněk Fabián and Igor Vajda


On bounded or unbounded intervals of the real line, we introduce classes of regular statistical families, called Johnson families because they are obtained using generalized Johnson transforms. We study in a rigorous manner the formerly introduced concept of core function of a distribution from a Johnson family, which is a modification of the well known score function and which in a one-to-one manner represents the distribution. Further, we study Johnson parametrized families obtained by Johnson transforms of location and scale families, where the location is replaced by a new parameter called Johnson location. We show that Johnson parametrized families contain many important statistical models. One form appropriately normalized $L_2$ distance of core functions of arbitrary distributions from Johnson families is used to define a core divergence of distributions. The core divergence of distributions from parametrized Johnson families is studied as a special case.