A new class of metric divergences on probability spaces and its applicability in statistics

The classI _f β, βε(0, ∞], off-divergences investigated in this paper is defined in terms of a class of entropies introduced by Arimoto (1971,Information and Control,19, 181–194). It contains the squared Hellinger distance (for β=1/2), the sumI(Q ₁‖(Q ₁+Q ₂)/2)+I(Q ₂‖(Q ₁+Q ₂)/2) of Kullback-Leibler divergences (for β=1) and half of the variation distance (for β=∞) and continuously extends the class of squared perimeter-type distances introduced by Österreicher (1996,Kybernetika,32, 389–393) (for βε (1, ∞]). It is shown that\((I_{f_\beta } (Q_1 ,Q_2 ))^{\min (\beta ,1/2)}\) are distances of probability distributionsQ ₁,Q ₂ for β ε (0, ∞). The applicability of\(I_{f_\beta }\)-divergences in statistics is also considered. In particular, it is shown that the\(I_{f_\beta }\)-projections of appropriate empirical distributions to regular families define distribution estimates which are in the case of an i.i.d. sample of size'n consistent. The order of consistency is investigated as well.