ON A CLASS OF PERIMETER-TYPE DISTANCES OF PROBABILITY-DISTRIBUTIONS

The class I-fp, p is an element of (1, infinity], of f-divergences inv estigated in this paper generalizes an f-divergence introduced by the author in [9] and applied there and by Reschenhofer and Bomze [11] in different areas of hypotheses testing. The main result of the present paper ensures that, for every p is an element of (1, infinity), the sq uare root of the corresponding divergence defines a distance on the se t of probability distributions. Thus it generalizes the respecting sta tement for p = 2 made in connection with Example 4 by Kafka, Osterreic her and Vincze in [6]. From the former literature on the subject the m aximal powers of f-divergences defining a distance are known for the s ubsequent classes. For the class of Hellinger-divergences given in ter ms of f((s))(u) = 1+u-(u(s)+u(1-s)), s is an element of (0, 1), alread y Csiszar and Fischer [3] have shown that the maximal power is min (s, 1-s). For the following two classes the maximal power coincides with their parameter. The class given in terms of f((alpha))(u) = \1-u(alph a)\(1/alpha), alpha is an element of (0, 1], was investigated by Boeke e [2]. The previous class and this one have the special case s = alpha = 1/2 in common. This famous case is attributed to Matusita [8]. The class given by phi(alpha)(u) = \1-u\(1/alpha)(1+u)(1-1/alpha), alpha i s an element of (0, 1], and investigated in [6], Example 3, contains t he wellknown special case alpha = 1/2 introduced by Vincze [13].