Nonparametric Functions Estimation Using Biased Data

Biased or weighted sampling frequently arises in reliability testing, biomedical survival analysis, and quality-control studies, where the observed data deviate systematically from the target population. This paper develops a unified framework for nonparametric estimation of probability density distribution, hazard rate, and regression functions when the data are subject to biased sampling. The proposed weighted kernel estimators adjust for biasing functions w(x), enabling asymptotically unbiased estimation under general sampling distortions. Comprehensive theoretical results are provided, including bias-variance decompositions, optimal bandwidth orders, and mean-squared error properties. Extensive numerical simulations and a real-data application to the Channing House dataset demonstrate the practical advantages and robustness of the proposed estimators compared with naïve approaches. The results confirm the method’s theoretical validity and its broad applicability in survival and reliability studies involving biased data.