A New Class of Metric Divergences on Probability Spaces and Its Applicability in Statistics
نویسنده
چکیده
A b s t r a c t . The class Ii0 , ~ C (0, c~], 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/~ = 1/2), the sum I(Q1 II (Q1 +Q2)/2) +I(Q~ II (Q1 +Q2)/2) of Kullback-Leibler divergences (for ~ = 1) and half of the variation distance (for/3 = c~) and continuously extends the class of squared perimeter-type distances introduced by ()sterreicher (1996, Kybernetika, 32, 389-393) (for j3 C (1,cx~]). It is shown that (IIo(Q1,Q2)) min(~'l/~) are distances of probability distributions Qt, Q2 for fl E (0, oo). The applicability of//0-divergences in statistics is also considered. In particular, it is shown that the/ /o-project ions 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.
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