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On one generalization of Chernov's distance
N. P. Salikhov Essential Administration of Information Systems
Abstract:
The variable
$\rho(\mathbf{p};A,B)$ is introduced to characterize, for a given vector
$\mathbf{p}$, the distance between finite sets
$A$ and
$B$ of vectors of probabilities of outcomes in polynomial schemes of trials having a common set of outcomes. In the case of singletons
$A=\{\mathbf{a}\}$,
$B=\{\mathbf{p}\}$ the value of
$\rho(\mathbf{p};A,B)$ coincides with the Chernov distance between
$\mathbf{p}$ and
$\mathbf{a}$. We indicate the probabilistic sense of the generalized Chernov distance
$\rho(\mathbf{p};A,B)$ and establish some of its properties. For distinguishing between
$m$ polynomial distributions
$(n,\mathbf{p}_1),\dots,(n,\mathbf{p}_m)$ we consider a Bayesian decision rule, where the proper distribution is found in
$k\in\{1,\dots,m-1\}$ most plausible variants. For this rule, we find explicit and asymptotic (as
$n\to\infty$) estimates of probabilities of errors depending on at most
$C_{m-1}^k$ generalized Chernov distances and, moreover, establish, in a sense, its optimality.
Keywords:
polynomial scheme of trials, Kullback–Leibler distance, Chernov distance, distinguishing between several simple hypotheses, Bayesian decision rule, estimates of probabilities of errors. Received: 14.01.1997
DOI:
10.4213/tvp1466
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