Context-sensitive hypothesis-testing and exponential families

We propose a number of concepts and properties related to ‘weighted’ statistical inference where the observed data are classified in accordance with a ‘value’ of a sample string. The motivation comes from the concepts of weighted information and weighted entropy that proved useful in industrial/microeconomic and medical statistics. We focus on applications relevant in hypothesis testing and an analysis of exponential families. Several notions, bounds and asymptotics are established, which generalize their counterparts well-known in standard statistical research. It includes Fisher information, Neyman–Pearson lemma, Stein–Sanov theorem, Pinsker's and Bretangnole–Huber bounds, Cramér–Rao and van Trees inequalities and Bhattacharyya, Bregman, Burbea-Rao, Chernoff, Kullback–Leibler, Rényi and Tsallis divergences.