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Lifetime Data Analysis

, Volume 13, Issue 1, pp 51–73 | Cite as

Nonparametric estimation of the mean function of a stochastic process with missing observations

  • X. Joan Hu
  • Stephen W. Lagakos
Article

Abstract

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right- censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.

Keywords

Censored survival data Discrete state process Panel data Repeated measures Weighted least squares estimation 

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Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of Statistics and Actuarial ScienceSimon Fraser UniversityBurnabyCanada
  2. 2.Department of BiostatisticsHarvard School of Public HealthBostonUSA

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