Nonparametric estimation of the mean function of a stochastic process with missing observations
- 126 Downloads
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.
KeywordsCensored survival data Discrete state process Panel data Repeated measures Weighted least squares estimation
Unable to display preview. Download preview PDF.
- Andersen PK, Borgan O, Gill RD, Keiding N (1992) Statistical models based on counting processes. Springer-Verlag, New YorkGoogle Scholar
- Barlow RE, Bartholomew DJ, Bremner JM, Brunk HD (1972) Statistical inference under order restrictions: the theory and applications of isotonic regression. Wiley, New YorkGoogle Scholar
- Diggle PJ, Liang KY, Zeger SL (1994) Analysis of longitudinal data. Clarendon Press, OxfordGoogle Scholar
- Gulick RM, Hu XJ, Fiscus SA, Courtney VF, Haubrich R, Cheng H, Scosta E, Lagakos SW, Swanstrom R, Freimuth W, Snyder S, Mills C, Fischl M, Pettinelli C, Katzenstein D (2000) Randomized study of saquinavir with ritonavir or nelfinavir together with delavirdine, adefovir or both in HIV-infected adults with virologic failure on indinavir: AIDS Clinical Trials Group (ACTG) Study 359. J Infect Dis 182:1375–1384CrossRefGoogle Scholar
- Gulick RM, Hu XJ, Fiscus SA, Courtney VF, Haubrich R, Cheng H, Scosta E, Lagakos SW, Swanstrom R, Freimuth W, Snyder S, Mills C, Fischl M, Pettinelli C, Katzenstein D (2002) Durability of response to treatment for antiretroviral-experienced subjects: 48 week results from AIDS Clinical Trials Group (ACTG) Study 359. J Infect Dis 186:626–633CrossRefGoogle Scholar
- Pollard D Empirical processes: theory and applications, Regional Conference Series in Probability and Statistics 2. Institute of Mathematical Statistics, Hayward, CAGoogle Scholar