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

The class I-fbeta, beta is an element of (0, infinity], of f-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 beta = 1/2), the sum I(Q(1)\\(Q(1) + Q(2))/2) +I(Q2\\(Q(1) + Q(2))/2) of Kullback-Leibler divergences (for beta = 1) and half of the variation distance (for beta = infinity) and continuously extends the class of squared perimeter-type distances introduced by Osterreicher (1996, Kybernetika, 32, 389-393) (for beta is an element of (1, infinity]). It is shown that (I-fbeta (Q(1),Q(2)))(min(beta,1/2)) are distances of probability distributions Q(1), Q(2) for beta is an element of (0, infinity). The applicability of I-fbeta-divergences in statistics is also considered. In particular, it is shown that the I-fbeta-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.