Asymptotic Normality of Wavelet Density and Regression Estimators Under Censored Ergodic Observations

Abstract

This paper develops a pointwise distributional theory for linear wavelet density and regression estimation from randomly right-censored observations exhibiting stationary ergodic dependence. In contrast to the prevailing literature, which typically relies on quantitative mixing conditions, our analysis is conducted under ergodicity alone, thereby encompassing substantially broader classes of dependent processes. We establish asymptotic normality for an oracle inverse-probability-weighted estimator based on the true censoring distribution and for its feasible counterpart obtained through Kaplan–Meier substitution. A central result shows that estimating the censoring distribution has no first-order effect on the limiting law, so that the feasible and oracle procedures are asymptotically equivalent. The proof strategy departs from conventional covariance inequalities and blocking arguments and instead combines a martingale-predictable decomposition with martingale central limit theory and ergodic convergence of conditional moments. The framework is further extended to a broad family of wavelet regression functionals involving transformed responses. To render the asymptotic theory directly usable for statistical inference, we introduce a randomly weighted procedure that consistently reproduces the limiting distribution of the feasible estimator. This yields asymptotically valid pointwise confidence intervals without requiring explicit estimation of the unknown asymptotic variance or the introduction of additional smoothing parameters. The scope of the theory includes several important non-mixing and long-range dependent models, while an extensive simulation study demonstrates the finite-sample accuracy and robustness of the proposed inferential methodology.