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

Area of Science:

• Biostatistics
• Statistical Inference
• Survival Analysis

Background:

• Estimating mean functions with incomplete data is challenging across diverse processes.
• Existing methods lack a unified theoretical framework for different response and observation types.

Purpose of the Study:

• To develop a general theoretical framework for nonparametric mean function estimation under incomplete observations.
• To unify estimation methods across various quantitative, survival, counting, and alternating binary processes.
• To accommodate general response-independent observation processes, including censoring and periodic sampling.

Main Methods:

• Proposed a general theoretical framework for nonparametric mean function estimation.
• Explored two criteria for estimator development: sample mean and Kaplan-Meier inspired approach.
• Investigated consistency and weak convergence to Gaussian processes under regularity conditions.

Main Results:

• Developed consistent estimators for mean functions and their covariance functions.
• Demonstrated that the proposed criteria yield familiar estimators for some cases and novel estimators for others.
• Illustrated the application of the framework using data from an AIDS clinical trial.

Conclusions:

• The general framework provides a unified approach to nonparametric mean function estimation with incomplete data.
• The proposed methods are statistically sound, offering consistent estimation and convergence properties.
• The framework has practical implications, leading to new estimators for complex data scenarios.