Open Access
April, 1969 Distinguishability of Probability Measures
Lloyd Fisher, John W. Van Ness
Ann. Math. Statist. 40(2): 381-392 (April, 1969). DOI: 10.1214/aoms/1177697702


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.


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Lloyd Fisher. John W. Van Ness. "Distinguishability of Probability Measures." Ann. Math. Statist. 40 (2) 381 - 392, April, 1969.


Published: April, 1969
First available in Project Euclid: 27 April 2007

zbMATH: 0174.22604
MathSciNet: MR247693
Digital Object Identifier: 10.1214/aoms/1177697702

Rights: Copyright © 1969 Institute of Mathematical Statistics

Vol.40 • No. 2 • April, 1969
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