Abstract:

The class $I_{f_p} , p\in (1,\infty] ,$ of $f$-divergences investigated 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 $p\in (1,\infty) ,$ the square root of the corresponding divergence defines a distance on the set of probability distributions. Thus it generalizes the respecting statement for $p=2$ made in connection with Example 4 by Kafka, Österreicher and Vincze in [6]. From the former literature on the subject the maximal powers of $f$-divergences defining a distance are known for the subsequent classes. For the class of Hellinger-divergences given in terms of $f^{(s)}(u) = 1+u- (u^s+u^{1-s}) , s\in (0,1) ,$ already Csiszár 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 $f_{(\alpha )}(u) = |1-u^{\alpha }|^{\frac{1}{\alpha }} , \alpha \in (0,1] ,$ was investigated by Boekee [2]. The previous class and this one have the special case $s = \alpha = \frac{1}{2}$ in common. This famous case is attributed to Matusita [8]. The class given by $\varphi _{\alpha }(u) = |1-u|^{\frac{1}{\alpha }} (1+u)^{1-{\frac{1}{\alpha }}} , \alpha \in (0,1] ,$ and investigated in [6], Example 3, contains the wellknown special case $\alpha ={1\over 2}$ introduced by Vincze [13].