Decompositions of information divergences: Recent development, open problems and applications

What is the optimal statistical decision? And how it is related to the statistical information theory? By trying to answer these difficult questions, we will illustrate the necessity of understanding of structure of information divergences. This may be understand in particular through deconvolutions, leading to an optimal statistical inference. We will illustrate deconvolution of information divergence in the exponential family, which will gave us an optimal tests (optimal in the sense of Bahadur (see [3, 4]). We discuss about the results on the exact density of the I-divergence in the exponential family with gamma distributed observations (see [27]). Since the considered I-divergence is related to the likelihood ratio (LR) statistics, we deal with the exact distribution of the likelihood ratio tests and discuss the optimality of such exact tests. The both tests, the exact LR test of the homogeneity and the exact LR test of the scale parameter, are asymptotically optimal in the Bahadur sense when the observations are distributed exponentially. We also discuss decompositions from a broader perspective. We recall relationship between f -divergence and statistical information in the sense of DeGroot, which was shown in [16]. We formulate an open problem of its generalization. Applications in reliability testing and hydrological prediction are mentioned.