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December, 1960 Asymptotic Rate of Discrimination for Markov Processes
Lambert H. Koopmans
Ann. Math. Statist. 31(4): 982-994 (December, 1960). DOI: 10.1214/aoms/1177705671


Simple hypotheses $H_P$ and $H_Q$, specifying two distinct positive transition densities $p(x \mid y)$ and $q \mid y)$ and initial densities $p(x)$ and $q(x)$ with respect to a finite Lebesque-Stieltjes measure, are assumed for a discrete time parameter Markov process. Let $R_n$ be the likelihood ratio static based on the first $n + 1$ observations of the process, and consider the class of sequences of likelihood ratio tests $T(a, \alpha) = \{\lbrack R_n > n^\alpha a\rbrack: n = 0, 1, 2, \cdots\}$ generated by letting $a$ and $\alpha$ vary over the real numbers. Under certain regularity assumptions on $K_t(x, y) = p^{1-t} (x \mid y)q^t (x \mid y)$ and the initial densities $p$ and $q$, the subclass of consistent sequences is determined, and the limiting rates at which the error probabilities tend to zero for tests in this subclass are found. A definition of the best asymptotic rate for distinguishing between $H_P$ and $H_Q$ is made for the class of consistent tests. This "asymptotic rate of discrimination" is evaluated and is shown to be attained by a certain subclass of these tests. Some applications and extensions of the theory to infinite Lebesgue-Stieltjes measures are given.


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Lambert H. Koopmans. "Asymptotic Rate of Discrimination for Markov Processes." Ann. Math. Statist. 31 (4) 982 - 994, December, 1960.


Published: December, 1960
First available in Project Euclid: 27 April 2007

zbMATH: 0096.12603
MathSciNet: MR119368
Digital Object Identifier: 10.1214/aoms/1177705671

Rights: Copyright © 1960 Institute of Mathematical Statistics

Vol.31 • No. 4 • December, 1960
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