TY - JOUR

T1 - Estimating the expected total number of events in a process

AU - Maller, Ross A

AU - Sun, Liuquan

AU - Zhou, Xian

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Age profile

KW - Censored observations

KW - Cumulative mean function

KW - Kaplan-Meier estimator

KW - Nelson-Aalen estimator

KW - Recurrent process

UR - http://www.scopus.com/inward/record.url?scp=0035998828&partnerID=8YFLogxK

U2 - 10.1198/016214502760047104

DO - 10.1198/016214502760047104

M3 - Article

SN - 0162-1459

VL - 97

SP - 577

EP - 589

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

IS - 458

ER -