Inverse-Probability-Weighted Wavelet Estimation of Regression Derivatives Under Missing-at-Random Responses for Stationary Ergodic Processes

Abstract

We consider the estimation of partial derivatives of multivariate regression-type functionals from incomplete observations generated by a discrete-time strictly stationary ergodic process. The response variable is subject to a missing-at-random (MAR) mechanism, whereas the covariates are fully observed. Building upon the complete-data wavelet methodology developed in Didi and Bouzebda (2025), we construct inverse-probability-weighted empirical wavelet estimators that compensate for the selection bias induced by missing responses. When the propensity score is unknown, a feasible estimator is obtained by replacing the oracle weights with a nonparametric Nadaraya–Watson estimator. The analysis is carried out under stationary ergodicity without imposing mixing assumptions. The estimation error is decomposed into three analytically distinct components: the deterministic multiresolution approximation error, the stochastic fluctuation of the oracle inverse-probability-weighted estimator, and the additional error arising from propensity score estimation. This decomposition makes it possible to isolate the respective effects of approximation, dependence, and missingness within a unified asymptotic framework. Under explicit assumptions on the multiresolution approximation, missingness mechanism, conditional density stabilization, moment conditions, and accuracy of the propensity estimator, we establish non-asymptotic integrated mean squared error bounds together with their asymptotic rates. We further prove almost-sure uniform consistency over compact subsets of the interior of the support and derive a pointwise central limit theorem for both the oracle and feasible estimators. The limiting variance explicitly reflects the information loss induced by inverse probability weighting, and for general orthogonal projection kernels is formulated under the corresponding dyadic-phase condition. The general methodology is specialized to the estimation of first- and second-order derivatives of ordinary regression functions. A finite-sample simulation study investigates the empirical behavior of the proposed estimators under stationary ergodic dependence and MAR missingness, examines the influence of both the wavelet resolution level and the propensity-score bandwidth, evaluates the finite-sample performance of the asymptotic confidence intervals, and compares the proposed procedure with oracle, complete-case, and competing nonparametric estimators. The numerical results are consistent with the theoretical analysis and illustrate the respective contributions of wavelet approximation, inverse probability weighting, and propensity score estimation to the overall estimation error. When the propensity score is identically equal to one, the proposed methodology reduces to the corresponding complete-data wavelet estimator.