Central Limit Theorem of the Recursive Estimate of Density Function Under Randomly Censored Data

Kernel density estimation for right-censored data has been extensively studied in the non-recursive setting, whereas recursive approaches adapted to censoring remain largely unexplored despite their considerable computational advantages in sequential data environments. In this paper, we introduce a recursive kernel density estimator for independent right-censored observations through a Kaplan-Meier weighting scheme. The proposed estimator can be updated incrementally as new observations become available, avoiding repeated re-computation of the entire estimator and substantially reducing memory and computational requirements. Under mild regularity conditions, we establish the asymptotic normality of the estimator and derive its asymptotic variance, which explicitly reflects the effect of the recursive weighting mechanism and the censoring process. We also construct asymptotic confidence intervals for the underlying density using a plug-in variance estimator. An extensive Monte Carlo study, including Gaussian, exponential, heavy-tailed, multimodal, contaminated, and severely censored scenarios, demonstrates that the proposed estimator achieves estimation accuracy comparable to that of the classical censored Parzen-Rosenblatt estimator while offering substantial computational gains. In particular, the recursive procedure remains stable under high censoring levels and exhibits excellent scalability for large and sequentially collected datasets. The proposed methodology provides an efficient and theoretically justified alternative for nonparametric density estimation under right censoring and is particularly suited to applications involving streaming data, such as survival analysis, reliability engineering, medical monitoring, and online forecasting.