{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T16:49:41Z","timestamp":1786121381837,"version":"build-2736575974"},"reference-count":56,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T00:00:00Z","timestamp":1785888000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2013Watson 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.<\/jats:p>","DOI":"10.3390\/e28080883","type":"journal-article","created":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T13:16:29Z","timestamp":1785935789000},"page":"883","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Inverse-Probability-Weighted Wavelet Estimation of Regression Derivatives Under Missing-at-Random Responses for Stationary Ergodic Processes"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7801-4945","authenticated-orcid":false,"given":"Salim","family":"Bouzebda","sequence":"first","affiliation":[{"name":"LMAC (Laboratory of Applied Mathematics of Compi\u00e8gne), Universit\u00e9 de Technologie de Compi\u00e8gne, 60200 Compi\u00e8gne, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7630-4604","authenticated-orcid":false,"given":"Sultana","family":"Didi","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Sciences, Qassim University, P.O. Box 6688, Buraydah 51452, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wand, M.P., and Jones, M.C. (1995). Kernel Smoothing, Chapman and Hall, Ltd.. Monographs on Statistics and Applied Probability.","DOI":"10.1007\/978-1-4899-4493-1"},{"key":"ref_2","unstructured":"Eggermont, P.P.B., and LaRiccia, V.N. (2009). Maximum Penalized Likelihood Estimation: Volume II: Regression, Springer. Springer Series in Statistics."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1111\/rssb.12111","article-title":"Non-parametric inference for density modes","volume":"78","author":"Genovese","year":"2016","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1930","DOI":"10.1162\/neco_a_01092","article-title":"Bias reduction and metric learning for nearest-neighbor estimation of Kullback-Leibler divergence","volume":"30","author":"Noh","year":"2018","journal-title":"Neural Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1134\/S0005117910020050","article-title":"Bandwidth selection in nonparametric estimator of density derivative by smoothed cross-validation method","volume":"71","author":"Dobrovidov","year":"2010","journal-title":"Autom. Remote Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3236","DOI":"10.1214\/08-AOS671","article-title":"On the path density of a gradient field","volume":"37","author":"Genovese","year":"2009","journal-title":"Ann. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1111\/j.2517-6161.1977.tb01635.x","article-title":"Applications of estimators of a density and its derivatives to certain statistical problems","volume":"39","author":"Singh","year":"1977","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1214\/aos\/1176343746","article-title":"Bounds for estimation of density functions and their derivatives","volume":"5","author":"Meyer","year":"1977","journal-title":"Ann. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1214\/aos\/1176344076","article-title":"Weak and strong uniform consistency of the kernel estimate of a density and its derivatives","volume":"6","author":"Silverman","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1080\/10485252.2021.1925668","article-title":"Uniform almost sure convergence and asymptotic distribution of the wavelet-based estimators of partial derivatives of multivariate density function under weak dependence","volume":"33","author":"Allaoui","year":"2021","journal-title":"J. Nonparametr. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1080\/10485252.2020.1834107","article-title":"Uniform convergence rate of the kernel regression estimator adaptive to intrinsic dimension in presence of censored data","volume":"32","author":"Bouzebda","year":"2020","journal-title":"J. Nonparametr. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"163","DOI":"10.3103\/S1066530715030011","article-title":"Multivariate wavelet density and regression estimators for stationary and ergodic continuous time processes: Asymptotic results","volume":"24","author":"Bouzebda","year":"2015","journal-title":"Math. Methods Stat."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ramsay, J.O., and Silverman, B.W. (2002). Applied Functional Data Analysis: Methods and Case Studies, Springer. Springer Series in Statistics.","DOI":"10.1007\/b98886"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ramsay, J.O., and Silverman, B.W. (2005). Functional Data Analysis, Springer. [2nd ed.]. Springer Series in Statistics.","DOI":"10.1007\/b98888"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107473","DOI":"10.1016\/j.csda.2022.107473","article-title":"A generalized correlated Cp criterion for derivative estimation with dependent errors","volume":"171","author":"Liu","year":"2022","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1287","DOI":"10.1080\/01621459.1993.10476410","article-title":"Confidence bands in nonparametric regression","volume":"88","author":"Eubank","year":"1993","journal-title":"J. Amer. Stat. Assoc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1080\/01621459.1995.10476630","article-title":"An effective bandwidth selector for local least squares regression","volume":"90","author":"Ruppert","year":"1995","journal-title":"J. Amer. Stat. Assoc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3954","DOI":"10.1016\/j.csda.2008.01.006","article-title":"SiZer analysis for the comparison of regression curves","volume":"52","author":"Park","year":"2008","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_19","first-page":"233","article-title":"On robust kernel estimation of derivatives of regression functions","volume":"12","author":"Gasser","year":"1985","journal-title":"Scand. J. Stat."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/S0378-3758(02)00152-0","article-title":"On the asymptotic normality of kernel regression estimators of the mode in the nonparametric random design model","volume":"115","author":"Ziegler","year":"2003","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/BF02481984","article-title":"Speed of convergence in nonparametric kernel estimation of a regression function and its derivatives","volume":"36","author":"Georgiev","year":"1984","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1023\/B:SISP.0000049092.55534.af","article-title":"General asymptotic confidence bands based on kernel-type function estimators","volume":"7","author":"Deheuvels","year":"2004","journal-title":"Stat. Inference Stoch. Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s13163-020-00368-6","article-title":"Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes","volume":"34","author":"Bouzebda","year":"2021","journal-title":"Rev. Mat. Complut."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"14807","DOI":"10.3934\/math.2024720","article-title":"Weak convergence of the conditional single index U-statistics for locally stationary functional time series","volume":"9","author":"Bouzebda","year":"2024","journal-title":"AIMS Math."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"H\u00e4rdle, W., Kerkyacharian, G., Picard, D., and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications, Springer. Lecture Notes in Statistics.","DOI":"10.1007\/978-1-4612-2222-4"},{"key":"ref_26","first-page":"91","article-title":"Nonparametric estimation of the derivatives of a density by the method of wavelets","volume":"28","year":"1996","journal-title":"Bull. Inf. Cybern."},{"key":"ref_27","first-page":"97","article-title":"Wavelet based estimation of the derivatives of a density with associated variables","volume":"27","author":"Chaubey","year":"2006","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_28","unstructured":"Madan, L.P. (2000). Nonparametric Estimation of Partial Derivatives of a Multivariate Probability Density by the Method of Wavelets. Asymptotics in Statistics and Probability, De Gruyter. Papers in Honor of George Gregory Roussas."},{"key":"ref_29","first-page":"31","article-title":"Nonparametric estimation of a multivariate probability density for mixing sequences by the method of wavelets","volume":"28","author":"Hosseinioun","year":"2011","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_30","unstructured":"Koshkin, G., and Vasil\u2019iev, V. (1997). An estimation of a multivariate density and its derivatives by weakly dependent observations. Statistics and Control of Stochastic Processes, World Scientific. The Liptser Festschrift; Papers from the Steklov Seminar Held in Moscow, Russia, 1995\u20131996."},{"key":"ref_31","first-page":"834","article-title":"Wavelet estimation for derivative of a density in the presence of additive noise","volume":"32","year":"2018","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Didi, S., and Bouzebda, S. (2025). Wavelet Estimation of Partial Derivatives in Multivariate Regression Under Discrete-Time Stationary Ergodic Processes. Mathematics, 13.","DOI":"10.3390\/math13101587"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Little, R.J.A., and Rubin, D.B. (2002). Statistical Analysis with Missing Data, Wiley-Interscience [John Wiley & Sons]. [2nd ed.]. Wiley Series in Probability and Statistics.","DOI":"10.1002\/9781119013563"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1214\/13-STS415","article-title":"What is meant by \u201cmissing at random\u201d?","volume":"28","author":"Seaman","year":"2013","journal-title":"Stat. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1093\/biomet\/asv035","article-title":"Clarifying missing at random and related definitions, and implications when coupled with exchangeability","volume":"102","author":"Mealli","year":"2015","journal-title":"Biometrika"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1214\/009053604000000166","article-title":"Missing at random, likelihood ignorability and model completeness","volume":"32","author":"Lu","year":"2004","journal-title":"Ann. Stat."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1093\/biomet\/asab002","article-title":"Missing at random: A stochastic process perspective","volume":"109","author":"Farewell","year":"2022","journal-title":"Biometrika"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bouzebda, S. (2026). Statistical Learning of Conditional Single-Index U-Processes Under Local Stationarity and Missing-at-Random Functional Responses. Mathematics, 14.","DOI":"10.3390\/math14122112"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Bouzebda, S. (2026). Advanced Statistical Learning: Limit Theorems for Nonparametric Conditional U-Statistics Smoothed by Asymmetric Kernels Under Missing-at-Random Sampling. Mathematics, 14.","DOI":"10.3390\/math14122110"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Bouzebda, S. (2026). Asymptotic Learning Theory for Conditional U\u2013Statistics Based on Delta Sequences Under Missing at Random Mechanisms. Mathematics, 14.","DOI":"10.3390\/math14111899"},{"key":"ref_41","unstructured":"Claeskens, G., and Hjort, N.L. (2008). Model Selection and Model Averaging, Cambridge University Press. Cambridge Series in Statistical and Probabilistic Mathematics."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1093\/biomet\/63.3.581","article-title":"Inference and missing data","volume":"63","author":"Rubin","year":"1976","journal-title":"Biometrika"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1214\/18-STS332IN","article-title":"Introduction to the special section on missing data","volume":"33","author":"Josse","year":"2018","journal-title":"Stat. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Little, R.J.A., and Rubin, D.B. (2019). Statistical Analysis with Missing Data, John Wiley & Sons. [3rd ed.]. Wiley Series in Probability and Statistics.","DOI":"10.1002\/9781119482260"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"834107","DOI":"10.1155\/2012\/834107","article-title":"General bootstrap for dual \u03d5-divergence estimates","volume":"2012","author":"Bouzebda","year":"2012","journal-title":"J. Probab. Stat."},{"key":"ref_46","unstructured":"Meyer, Y. (1992). Wavelets and Operators, Cambridge University Press. Cambridge Studies in Advanced Mathematics."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Triebel, H. (1983). Theory of Function Spaces, Birkh\u00e4user. Monographs in Mathematics.","DOI":"10.1007\/978-3-0346-0416-1"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1214\/aop\/1176997023","article-title":"Distribution function inequalities for martingales","volume":"1","author":"Burkholder","year":"1973","journal-title":"Ann. Probab."},{"key":"ref_49","unstructured":"de la Pe\u00f1a, V.H., and Gin\u00e9, E. (1999). From dependence to independence, Randomly stopped processes. U-statistics and processes. Martingales and beyond. Decoupling, Springer. Probability and its Applications (New York)."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1080\/10485250008832809","article-title":"Wavelet-based estimation of multivariate regression functions in Besov spaces","volume":"12","author":"Masry","year":"2000","journal-title":"J. Nonparametr. Stat."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2261","DOI":"10.1016\/j.jspi.2008.10.017","article-title":"Lower bound for estimation of Sobolev densities of order less 12","volume":"139","author":"Efromovich","year":"2009","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"DeVore, R.A., and Lorentz, G.G. (1993). Constructive Approximation, Springer. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].","DOI":"10.1007\/978-3-662-02888-9"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"455","DOI":"10.4171\/rmi\/463","article-title":"Superposition operators and functions of bounded p-variation","volume":"22","author":"Bourdaud","year":"2006","journal-title":"Rev. Mat. Iberoam."},{"key":"ref_54","unstructured":"Peetre, J. (1976). New Thoughts on Besov Spaces, Duke University, Mathematics Department. Duke University Mathematics Series, No. 1."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Dudley, R.M., and Norvai\u0161a, R. (1999). Differentiability of Six Operators on Nonsmooth Functions and p-Variation, Springer. Lecture Notes in Mathematics; With the Collaboration of Jinghua Qian.","DOI":"10.1007\/BFb0100744"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1007\/s12220-010-9150-3","article-title":"Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds","volume":"21","author":"Geller","year":"2011","journal-title":"J. Geom. Anal."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/883\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T04:12:12Z","timestamp":1786075932000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/883"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,5]]},"references-count":56,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["e28080883"],"URL":"https:\/\/doi.org\/10.3390\/e28080883","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,5]]}}}