On the Equivalence of $f$-Divergence Balls and Density Bands in Robust Detection

The paper deals with minimax optimal statistical tests for two composite hypotheses, where each hypothesis is defined by a nonparametric uncertainty set of feasible distributions. It is shown that for every pair of uncertainty sets of the $f$-divergence-ball type, a pair of uncertainty sets of the density-band type can be constructed, which is equivalent in the sense that it admits the same pair of least favorable distributions. This result implies that robust tests under $f$-divergence-ball uncertainty, which are typically only minimax optimal for the single sample case, are also fixed sample size minimax optimal with respect to the equivalent density-band uncertainty sets.