TRACKING A RANDOM WALK FIRST-PASSAGE TIME THROUGH NOISY OBSERVATIONS
Given a Gaussian random walk (or a Wiener process), possibly with drift, observed through noise, we consider the problem of estimating its first-passage time τ l of a given level l with a stopping time η defined over the noisy observation process. Main results are upper and lower bounds on the minim...
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| Veröffentlicht in: | The Annals of applied probability 2012-10, Vol.22 (5), p.1860-1879 |
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| container_end_page | 1879 |
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| container_issue | 5 |
| container_start_page | 1860 |
| container_title | The Annals of applied probability |
| container_volume | 22 |
| creator | Burnashev, Marat V. Tchamkerten, Aslan |
| description | Given a Gaussian random walk (or a Wiener process), possibly with drift, observed through noise, we consider the problem of estimating its first-passage time τ l of a given level l with a stopping time η defined over the noisy observation process. Main results are upper and lower bounds on the minimum mean absolute deviation infη E|η —τ l which become tight as l →∞. Interestingly, in this regime the estimation error does not get smaller if we allow η to be an arbitrary function of the entire observation process, not necessarily a stopping time. In the particular case where there is no drift, we show that it is impossible to track τ l : infη Elη — τ l | p = ∞ for any l > 0 and p ≥ 1/2. |
| doi_str_mv | 10.1214/11-AAP815 |
| format | Article |
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Main results are upper and lower bounds on the minimum mean absolute deviation infη E|η —τ l which become tight as l →∞. Interestingly, in this regime the estimation error does not get smaller if we allow η to be an arbitrary function of the entire observation process, not necessarily a stopping time. 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| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; Project Euclid Complete |
| subjects | 60G40 62L10 Errors Estimating techniques Estimators Integers Mathematical constants Mathematical functions Mean absolute deviation Normal distribution Optimal stopping quickest decision Random variables Random walk Random walk theory sequential analysis Statistics Stopping distances |
| title | TRACKING A RANDOM WALK FIRST-PASSAGE TIME THROUGH NOISY OBSERVATIONS |
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