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A simple and effective scanning rule for a multi-channel system. (English) Zbl 0855.62065

Summary: We consider a multichannel system in which one apparatus makes a sequence of observations, one at a time. By means of scanning, i.e. selecting a channel to be analyzed at any instant and deciding to stop at some stage, it is required to determine the channel in which there is the signal with prescribed constraints on error probabilities. A simple scanning rule, based on a cyclic application of a sequential probability ratio test (SPRT) is proposed for this problem. It is proved that in the case of Brownian motion, the expected scanning time of this rule is equal to the one of the optimal scanning rule (which is known only for this case). The simple structure of this rule permits to obtain corrected Brownian approximations for its characteristics in the case of exponentional families of distributions. The same procedure is used in multichannel change point problems.

MSC:

62L10 Sequential statistical analysis
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References:

[1] Bassevile M, Nikiforov IV (1993) Detection of abrupt changes: Theory and applications. PTR Prentice-Hall New Jersey
[2] Brodsky BE, Darkhovsky BS (1993) Nonparametric methods in change-point problems. Kluwer Academic Dordrecht
[3] Dragalin V (1989) Asymptotic solution of a problem of signal searching in multi-channel system. Mat Issled (Academy of Sciences of Moldova) 109:15–35 (In Russian)
[4] Dragalin V (1994) A multi-channel change point problem. Proceedings of 3rd Umeå-Würzburg Conference in Statistics. Umeå University
[5] Lotov VI (1987) Asymptotic expansions in the sequential probability ratio test. Theory Probab App 32:62–72 · Zbl 0622.62081
[6] Pollak M, Siegmund D (1985) A diffusion process and its application to detecting a change in the drift of Brownian motion. Biometrika 72:267–280 · Zbl 0571.60084 · doi:10.1093/biomet/72.2.267
[7] Shiryaev AN (1963) On the detection of disorder in a manufacturing process. Theory Probab Appl 8:247–265, 402–413 · Zbl 0279.90011 · doi:10.1137/1108029
[8] Shiryaev AN (1964) On the theory of decision function and control by observation from incomplete data. Trans Third Prague Conf Inform Theory Prague 657–687 (In Russian)
[9] Siegmund D (1985) Sequential analysis: Tests and confidence intervals. Springer Berlin · Zbl 0573.62071
[10] Srivastava MS, Wu Y (1993) Comparison of EWMA, CUSUM and Shiryayev-Roberts procedures for detecting a shift in the mean. Ann Statist 21:645–670 · Zbl 0816.62068 · doi:10.1214/aos/1176349142
[11] Woodroofe M (1982) Nonlinear renewal theory in sequential analysis. SIAM Philadelphia
[12] Zigangirov KSh (1966) On a problem in optimal scanning. Theory Probab Appl 11:294–298 · Zbl 0156.38903 · doi:10.1137/1111025
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