Abstract
Given a Gaussian random walk X with drift, we consider the problem of estimating its first-passage time τ A for a given level A from an observation process Y correlated to X. Estimators may be any stopping times η with respect to the observation process Y. Two cases of the process Y are considered: a noisy version of X and a process X with delay d. For a given loss function f(x), in both cases we find exact asymptotics of the minimal possible risk E f((η − τ A )/r) as A, d → ∞, where r is a normalizing coefficient. The results are extended to the corresponding continuous-time setting where X and Y are Brownian motions with drift.
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Original Russian Text © M.V. Burnashev, A. Tchamkerten, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 2, pp. 65–78.
Supported in part by the Russian Foundation for Basic Research (project no. 12-01-00905-a).
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Burnashev, M.V., Tchamkerten, A. Sequential estimation of a threshold crossing time for a Gaussian random walk through correlated observations. Probl Inf Transm 48, 142–153 (2012). https://doi.org/10.1134/S0032946012020044
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DOI: https://doi.org/10.1134/S0032946012020044