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September, 1985 Consistent Estimation in Partially Observed Random Walks
Peter Guttorp, Andrew F. Siegel
Ann. Statist. 13(3): 958-969 (September, 1985). DOI: 10.1214/aos/1176349649


If a discrete-time random walk, consisting of the sum of independent increments from an unknown underlying distribution, is observed at every time instant then it is clear that the underlying distribution can be consistently estimated. If, however, we are restricted to observing only a subset of the times, and if this subset is too sparse, then a central limit effect takes over and only two moments can be consistently estimated. We show that divergence of the sum of the reciprocals of the observation time-intervals is a necessary and sufficient condition to permit consistent estimation of the third moment. Corresponding conditions permit consistent estimation of moments of higher order. An explicit consistent estimator for the distribution itself is presented when all moments can be consistently estimated and the distribution is determined by its moments.


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Peter Guttorp. Andrew F. Siegel. "Consistent Estimation in Partially Observed Random Walks." Ann. Statist. 13 (3) 958 - 969, September, 1985.


Published: September, 1985
First available in Project Euclid: 12 April 2007

zbMATH: 0588.62058
MathSciNet: MR803751
Digital Object Identifier: 10.1214/aos/1176349649

Primary: 62G99

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 3 • September, 1985
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