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.
We establish an asymptotic theory for the Jones inverse-weighted kernel density estimator when length-biased observations form a strictly stationary short-range dependent sequence. The statistical difficulty is intrinsically composite: reciprocal weighting is singular at the origin, the normalizing mean is estimated from the same dependent sample, kernel localization shrinks with the bandwidth, and the centered summands form a row-wise stationary triangular array whose envelope diverges at rate hn−1. Under a non-negative compactly supported Lipschitz kernel, an inverse-moment condition, geometric α-mixing, local regularity of the target density, and uniform local bounds on lagged bivariate densities, we prove strong uniform consistency on compact subsets of (0,∞) and, separately, the uniform stochastic bound OP{hn2+(logn/(nhn))1/2}. A covariance-localization argument shows that the scaled serial-covariance contribution is O{hnlog(1/hn)}=o(1), so the first-order pointwise variance coincides with that of the corresponding independent length-biased estimator. Pointwise and finite-dimensional Gaussian limits are obtained by an explicit big-block/small-block argument with off-diagonal covariance control. The ratio normalization is treated directly: its variance contribution, its product with the localized fluctuation, and its cross-covariance with that fluctuation are all negligible at the nhn scale. We further derive first-order AMSE and AMISE criteria, their oracle bandwidths, and feasible pointwise studentization under undersmoothing. The numerical study separates oracle from data-driven bandwidth selection, evaluates full-ratio HAC and moving-block corrections, examines a Frank-copula Markov robustness design, and benchmarks the Jones estimator against an alternative length-biased estimator. The simulations support the first-order theory while demonstrating that persistent short-range dependence can remain consequential for finite-sample uncertainty.
Salim Bouzebda, S. Didi· Symmetry· 0 citations
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