Core functions and core divergences of regular distributions

summary: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