{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T07:42:17Z","timestamp":1769067737081,"version":"3.49.0"},"reference-count":35,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T00:00:00Z","timestamp":1637971200000},"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>This paper focuses on the adaptive spline (A-spline) fitting of the semiparametric regression model to time series data with right-censored observations. Typically, there are two main problems that need to be solved in such a case: dealing with censored data and obtaining a proper A-spline estimator for the components of the semiparametric model. The first problem is traditionally solved by the synthetic data approach based on the Kaplan\u2013Meier estimator. In practice, although the synthetic data technique is one of the most widely used solutions for right-censored observations, the transformed data\u2019s structure is distorted, especially for heavily censored datasets, due to the nature of the approach. In this paper, we introduced a modified semiparametric estimator based on the A-spline approach to overcome data irregularity with minimum information loss and to resolve the second problem described above. In addition, the semiparametric B-spline estimator was used as a benchmark method to gauge the success of the A-spline estimator. To this end, a detailed Monte Carlo simulation study and a real data sample were carried out to evaluate the performance of the proposed estimator and to make a practical comparison.<\/jats:p>","DOI":"10.3390\/e23121586","type":"journal-article","created":{"date-parts":[[2021,11,29]],"date-time":"2021-11-29T05:23:02Z","timestamp":1638163382000},"page":"1586","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Right-Censored Time Series Modeling by Modified Semi-Parametric A-Spline Estimator"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8393-1270","authenticated-orcid":false,"given":"Dursun","family":"Ayd\u0131n","sequence":"first","affiliation":[{"name":"Department of Statistics, Faculty of Science, Mugla Sitki Kocman University, Kotekli 48000, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Syed Ejaz","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, Brock University, 1812 Sir Isaac Brock Way, St. Catharines, ON L2S 3A1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9871-4700","authenticated-orcid":false,"given":"Ersin","family":"Y\u0131lmaz","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, Mugla Sitki Kocman University, Kotekli 48000, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1002\/cjs.5550350113","article-title":"Censored time series analysis with autoregressive moving average models","volume":"35","author":"Park","year":"2007","journal-title":"Can. J. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s10614-020-10010-8","article-title":"Censored nonparametric time-series analysis with autoregressive error models","volume":"58","author":"Aydin","year":"2020","journal-title":"Comput. Econ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1111\/j.0006-341X.2001.00022.x","article-title":"Multiple imputation for multivariate data with missing and below-threshold measurements: Time-series concentrations of pollutants in the arctic","volume":"57","author":"Hopke","year":"2001","journal-title":"Biometrics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1111\/j.1467-9469.2007.00586.x","article-title":"Non-parametric Regression with Dependent Censored Data","volume":"35","author":"Ghouch","year":"2008","journal-title":"Scand. J. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Koul, H., Susarla, V., and Van Ryzin, J. (1981). Regression Analysis with Randomly Right-Censored Data. Ann. Stat., 1276\u20131285.","DOI":"10.1214\/aos\/1176345644"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"301","DOI":"10.2307\/2336144","article-title":"Linear models, random censoring and synthetic data","volume":"74","author":"Leurgans","year":"1987","journal-title":"Biometrika"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1002","DOI":"10.1214\/aos\/1176348667","article-title":"Asymptotic Normality of the \u2018Synthetic Data\u2019 Regression Estimator for Censored Survival Data","volume":"20","author":"Zhou","year":"1992","journal-title":"Ann. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1111\/j.1368-423X.2009.00278.x","article-title":"Non-parametric regression with a latent time series","volume":"12","author":"Linton","year":"2010","journal-title":"Econom. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2601","DOI":"10.1214\/12-AOS1043","article-title":"Nonparametric regression for locally stationary time series","volume":"40","author":"Vogt","year":"2012","journal-title":"Ann. Stat."},{"key":"ref_10","unstructured":"Gao, J. (2007). Nonlinear Time Series: Semiparametric and Nonparametric Methods, CRC Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.jeconom.2012.07.001","article-title":"Semiparametric trending panel data models with cross-sectional dependence","volume":"171","author":"Chen","year":"2012","journal-title":"J. Econom."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1080\/01621459.1986.10478274","article-title":"Semiparametric Estimates of the Relation between Weather and Electricity Sales","volume":"80","author":"Engle","year":"1986","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"H\u00e4rdle, W. (1990). Applied Nonparametric Regression (No. 19), Cambridge University Press.","DOI":"10.1017\/CCOL0521382483"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Green, P.J., and Silverman, B.W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach, CRC Press.","DOI":"10.1007\/978-1-4899-4473-3"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ruppert, D., Wand, M.P., and Carroll, R.J. (2003). Semiparametric Regression (No. 12), Cambridge University Press.","DOI":"10.1017\/CBO9780511755453"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1524\/stnd.2008.0919","article-title":"On nonparametric estimation of the regression function under random censorship model","volume":"26","author":"Guessoum","year":"2009","journal-title":"Stat. Decis."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1470","DOI":"10.1080\/00949655.2018.1439032","article-title":"Modified estimators in semiparametric regression models with right-censored data","volume":"88","author":"Aydin","year":"2018","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1080\/01621459.1958.10501452","article-title":"Nonparametric Estimation from Incomplete Observations","volume":"53","author":"Kaplan","year":"1958","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1050","DOI":"10.1214\/aos\/1069362738","article-title":"On the rate of uniform convergence of the product-limit estimator: Strong and weak laws","volume":"25","author":"Chen","year":"1997","journal-title":"Ann. Stat."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1214\/aop\/1176990943","article-title":"Functional laws of the iterated logarithm for the product-limit estimator of a distribution function under random censorship or truncation","volume":"18","author":"Gu","year":"1990","journal-title":"Ann. Probab."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"De Boor, C. (1978). A Practical Guide to Splines, Springer.","DOI":"10.1007\/978-1-4612-6333-3"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Lyche, T., Manni, C., and Speleers, H. (2018). Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, Springer.","DOI":"10.1007\/978-3-319-94911-6_1"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Jin, H., Guan, Y., and Yao, L. (2017). Minimum entropy active fault tolerant control of the non-Gaussian stochastic distribution system subjected to mean constraint. Entropy, 19.","DOI":"10.3390\/e19050218"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Havrylenko, Y., Kholodniak, Y., Halko, S., Vershkov, O., Miroshnyk, O., Suprun, O., Dereza, O., Shchur, T., and \u015arutek, M. (2021). Representation of a Monotone Curve by a Contour with Regular Change in Curvature. Entropy, 23.","DOI":"10.3390\/e23070923"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1146","DOI":"10.1093\/bioinformatics\/bti148","article-title":"Quantile smoothing of array CGH data","volume":"21","author":"Eilers","year":"2005","journal-title":"Bioinformatics"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ahmed, S.E., Ayd\u0131n, D., and Y\u0131lmaz, E. (2020). Imputation Method Based on Sliding Window for Right-Censored Data. International Conference on Management Science and Engineering Management, Springer.","DOI":"10.1007\/978-3-030-49829-0_32"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Rippe, R.C., Meulman, J.J., and Eilers, P.H. (2012). Visualization of genomic changes by segmented smoothing using an L0 penalty. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0038230"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Frommlet, F., and Nuel, G. (2016). An adaptive ridge procedure for l 0 regularization. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0148620"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1111\/1467-9868.00125","article-title":"Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion","volume":"60","author":"Hurvich","year":"1998","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1214\/009053607000000604","article-title":"Variable selection in semiparametric regression modeling","volume":"36","author":"Li","year":"2008","journal-title":"Ann. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1214\/ss\/1038425655","article-title":"Flexible smoothing with B-splines and penalties","volume":"11","author":"Eilers","year":"1996","journal-title":"Stat. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1080\/00401706.1993.10485033","article-title":"A statistical view of some chemometrics regression tools","volume":"35","author":"Frank","year":"1993","journal-title":"Technometrics"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1356","DOI":"10.1214\/aos\/1015957397","article-title":"Asymptotics for lasso-type estimators","volume":"28","author":"Fu","year":"2000","journal-title":"Ann. Stat."},{"key":"ref_34","first-page":"1100","article-title":"Cox\u2019s regression model for counting processes: Large sample study","volume":"10","author":"Anderson","year":"1982","journal-title":"Ann. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1017\/S0266466600004394","article-title":"Asymptotics for least absolute deviation regression estimators","volume":"7","author":"Pollard","year":"1991","journal-title":"Econom. 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