Distinguishability of Probability Measures

Abstract

Independent identically distributed observations, $X_1, X_2, \cdots$, are taken sequentially. All that is known a priori about their common probability measure, $P$, is that it is a member of a given (at most countable) family, $\pi = \{P_n\}^\infty_{n=1}$, of such measures. At some time, depending only on the observed data and the tolerable probability of error, one wants to stop and decide which $P_k$ nature has chosen. Two sampling situations are considered, with and without error, as well as two stopping time requirements, uniformly (over $\pi$) bounded and $P_k$-dependent. Necessary and/or sufficient conditions for the distinguishability of the measures in $\pi$ in terms of a variety of measure metrics are obtained. The Levy-Prokhorov metric proves to be particularly relevant.