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Title: Estimating the expected total number of events in a process
Authors: Maller, RA
Sun, L
Zhou, X
Keywords: Age profile
Censored observations
Cumulative mean function
Kaplan-Meier estimator
Nelson-Aalen estimator
Recurrent process
Issue Date: 2002
Publisher: Amer Statistical Assoc
Source: Journal of the American statistical association, 2002, v. 97, no. 458, p. 577-589 How to cite?
Journal: Journal of the American Statistical Association 
Abstract: We consider estimation of the cumulative mean function of a process recurring in time, such as the numbers of arrests or migrations accrued by an individual, as a function of their age. We call this the age profile of a series of events. In some situations we can expect a finite value for the total number of events experienced by an individual, for example, when the distribution of the interevent times is improper, so that the process may cease at a finite time with positive probability. We propose and analyze a new estimator, constructed from Kaplan-Meier (KM) estimators of the interevent time distributions, for such an age profile, and compare it with the Nelson-Aalen (NA) estimator of the cumulative mean function of a process. The KM estimator is proved to be uniformly consistent for the age profile if and only if follow-up in the sample is sufficient in a sense that is manifested in practice by the leveling off of the profiles at their right-side ends, and is asymptotically normally distributed around its true value under mild conditions. Simulation results suggest that it is generally a better estimator than the NA estimator for the total number of events. It also appears to be more stable when applied to real data examples. For accurate estimation, it seems to be important to select a cohort of individuals whose ages at the first event are as similar as possible. The estimators are illustrated on some time-to-arrest data.
ISSN: 0162-1459
DOI: 10.1198/016214502760047104
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