Efficient sieve estimation of semiparametric probit model with doubly censored data

Lanxin Cui; Shishun Zhao; Shuwei Li; Lanxin Cui; Shishun Zhao; Shuwei Li

doi:10.3934/math.20251220

AIMS Mathematics

2025, Volume 10, Issue 11: 27755-27774. doi: 10.3934/math.20251220

Previous Article Next Article

Research article

Efficient sieve estimation of semiparametric probit model with doubly censored data

1.
School of Mathematics, Jilin University, Changchun 130000, China
2.
School of Economics and Statistics, Guangzhou University, Guangzhou 510000, China

Received: 23 September 2025 Revised: 13 November 2025 Accepted: 20 November 2025 Published: 27 November 2025
MSC : 62N02, 62G08

Semiparametric probit model serves as a valuable alternative to the popular proportional hazards/odds model in survival analysis partly due to the use of a standard normal distributed random error. This feature can facilitate developing an efficient inference and may render a better fit for the real world data than other models. In this work, we concern regression analysis of doubly censored data with a spline-based probit regression model and provide an efficient maximum likelihood estimation procedure. A novel and reliable expectation-maximization algorithm is proposed to identify the sieve estimator. Asymptotic properties of the proposed estimator are established. Simulation studies suggest that the proposed method works well in finite samples and obviously outperforms the direct sieve maximum likelihood method, which is accomplished with some existing optimization algorithm in the software. An application to a real data set is also provided.
- double censoring,
- maximum likelihood estimation,
- monotone splines,
- semiparametric model,
- survival analysis
Citation: Lanxin Cui, Shishun Zhao, Shuwei Li. Efficient sieve estimation of semiparametric probit model with doubly censored data[J]. AIMS Mathematics, 2025, 10(11): 27755-27774. doi: 10.3934/math.20251220

Related Papers:

Abstract

Semiparametric probit model serves as a valuable alternative to the popular proportional hazards/odds model in survival analysis partly due to the use of a standard normal distributed random error. This feature can facilitate developing an efficient inference and may render a better fit for the real world data than other models. In this work, we concern regression analysis of doubly censored data with a spline-based probit regression model and provide an efficient maximum likelihood estimation procedure. A novel and reliable expectation-maximization algorithm is proposed to identify the sieve estimator. Asymptotic properties of the proposed estimator are established. Simulation studies suggest that the proposed method works well in finite samples and obviously outperforms the direct sieve maximum likelihood method, which is accomplished with some existing optimization algorithm in the software. An application to a real data set is also provided.

References

[1]	E. A. Gehan, A generalized two-sample Wilcoxon test for doubly censored data, Biometrika, 52 (1965), 650–653. https://doi.org/10.2307/2333721 doi: 10.2307/2333721
[2]	B. W. Turnbull, Nonparametric estimation of a survivorship function with doubly censored data, J. Amer. Stat. Assoc., 69 (1974), 169–173. https://doi.org/10.1080/01621459.1974.10480146 doi: 10.1080/01621459.1974.10480146
[3]	S. Li, T. Hu, P. Wang, J. Sun, A class of semiparametric transformation models for doubly censored failure time data, Scandinavian J. Stat., 45 (2018), 682–698. https://doi.org/10.1111/sjos.12319 doi: 10.1111/sjos.12319
[4]	M. N. Chang, Weak convergence of a self-consistent estimator of the survival function with doubly censored data, Ann. Stat., 18 (1990), 391–404.
[5]	M. G. Gu, C. H. Zhang, Asymptotic properties of self-consistent estimators based on doubly censored data, Ann. Stat., 21 (1993), 611–624.
[6]	M. D. Hughes, Analysis and design issues for studies using censored biomarker measurements with an example of viral load measurements in HIV clinical trials, Stat. Med., 19 (2000), 3171–3191. https://doi.org/10.1002/1097-0258(20001215)19:23<3171::AID-SIM619>3.0.CO;2-T doi: 10.1002/1097-0258(20001215)19:23<3171::AID-SIM619>3.0.CO;2-T
[7]	C. H. Zhang, X. Li, Linear regression with doubly censored data, Ann. Stat., 24 (1996), 2720–2743. https://doi.org/10.1214/aos/1032181177 doi: 10.1214/aos/1032181177
[8]	J. J. Ren, M. Gu, Regression M-estimators with doubly censored data, Ann. Stat., 25 (1997), 2638–2664. https://doi.org/10.1214/aos/1030741089 doi: 10.1214/aos/1030741089
[9]	Y. Kim, B. Kim, W. Jang, Asymptotic properties of the maximum likelihood estimator for the proportional hazards model with doubly censored data, J. Multivar. Anal., 101 (2010), 1339–1351. https://doi.org/10.1016/j.jmva.2010.01.010 doi: 10.1016/j.jmva.2010.01.010
[10]	P. S. Shen, The Cox-Aalen model for doubly censored data, Commun. Stat.-Theory Meth., 51 (2022), 8075–8092. https://doi.org/10.1080/03610926.2021.1887241 doi: 10.1080/03610926.2021.1887241
[11]	T. Cai, S. Cheng, Semiparametric regression analysis for doubly censored data, Biometrika, 91 (2004), 277–290. https://doi.org/10.1093/biomet/91.2.277 doi: 10.1093/biomet/91.2.277
[12]	S. Ji, L. Peng, Y. Cheng, H. Lai, Quantile regression for doubly censored data, Biometrics, 68 (2012), 101–112. https://doi.org/10.1111/j.1541-0420.2011.01667.x doi: 10.1111/j.1541-0420.2011.01667.x
[13]	S. Li, T. Hu, T. Tong, J. Sun, Semiparametric regression analysis of multivariate doubly censored data, Stat. Model., 20 (2020), 502–526. https://doi.org/10.1177/1471082X19859 doi: 10.1177/1471082X19859
[14]	Y. Deng, S. Li, L. Sun, X. Song, Semiparametric probit regression model with general interval-censored failure time data, J. Comput. Graphical Stat., 33 (2024), 1413–1423. https://doi.org/10.1080/10618600.2024.2330523 doi: 10.1080/10618600.2024.2330523
[15]	X. Lin, L. Wang, A semiparametric probit model for case 2 interval-censored failure time data, Stat. Med., 29 (2010), 972–981. https://doi.org/10.1002/sim.3832 doi: 10.1002/sim.3832
[16]	X. Lin, L. Wang, Bayesian proportional odds models for analyzing current status data: univariate, clustered, and multivariate, Commun. Stat.-Simul. Comput., 40 (2011), 1171–1181. https://doi.org/10.1080/03610918.2011.566971 doi: 10.1080/03610918.2011.566971
[17]	H. Liu, J. Qin, Semiparametric probit models with univariate and bivariate current-status data, Biometrics, 74 (2018), 68–76. https://doi.org/10.1111/biom.12726 doi: 10.1111/biom.12726
[18]	M. Du, T. Hu, J. Sun, Semiparametric probit model for informative current status data, Stat. Med., 38 (2019), 2219–2227. https://doi.org/10.1002/sim.8106 doi: 10.1002/sim.8106
[19]	L. Fang, S. Li, L. Sun, X. Song, Semiparametric probit regression model with misclassified current status data, Stat. Med., 42 (2023), 4440–4457. https://doi.org/10.1002/sim.9869 doi: 10.1002/sim.9869
[20]	L. Fang, T. Hu, S. Li, L. Wang, C. S. McMahan, J. M. Tebbs, Probit time-to-event regression for misclassified grouptesting data, Stat. Sinica, 37 (2024), 1–25.
[21]	Y. T. Huang, T. Cai, Mediation analysis for survival data using semiparametric probit models, Biometrics, 72 (2016), 563–574. https://doi.org/10.1111/biom.12445 doi: 10.1111/biom.12445
[22]	J. Ramsay, Monotone regression splines in action, Stat. Sci., 3 (1988), 425–441.
[23]	X. Chen, T. Hu, J. Sun, Sieve maximum likelihood estimation for the proportional hazards model under informative censoring, Comput. Stat. Data Anal., 112 (2017), 224–234. https://doi.org/10.1016/j.csda.2017.03.006 doi: 10.1016/j.csda.2017.03.006
[24]	J. Huang, Asymptotic properties of nonparametric estimation based on partly interval-censored data, Stat. Sinica, 9 (1999), 501–519.
[25]	D. Zeng, F. Gao, D. Y. Lin, Maximum likelihood estimation for semiparametric regression models with multivariate interval-censored data, Biometrika, 104 (2017), 505–525. https://doi.org/10.1093/biomet/asx029 doi: 10.1093/biomet/asx029
[26]	J. Zhou, J. Zhang, W. Lu, Computationally efficient estimation for the generalized odds rate mixture cure model with interval-censored data, J. Comput. Graphical Stat., 27 (2018), 48–58. https://doi.org/10.1080/10618600.2017.1349665 doi: 10.1080/10618600.2017.1349665
[27]	S. Li, T. Hu, P. Wang, J. Sun, Regression analysis of current status data in the presence of dependent censoring with applications to tumorigenicity experiments, Comput. Stat. Data Anal., 110 (2017), 75–86. https://doi.org/10.1016/j.csda.2016.12.011 doi: 10.1016/j.csda.2016.12.011
[28]	L. Ma, T. Hu, J. Sun, Sieve maximum likelihood regression analysis of dependent current status data, Biometrika, 102 (2015), 731–738. https://doi.org/10.1093/biomet/asv020 doi: 10.1093/biomet/asv020
[29]	D. Zeng, Q. Chen, J. G. Ibrahim, Gamma frailty transformation models for multivariate survival times, Biometrika, 96 (2009), 277–291. https://doi.org/10.1093/biomet/asp008 doi: 10.1093/biomet/asp008
[30]	A. W. Van Der Vaart, J. A. Wellner, Weak convergence, In: Weak Convergence and Empirical Processes, 1996, 16–28.
[31]	X. Shen, W. H. Wong, Convergence rate of sieve estimates, Ann. Stat., 22 (1994), 580–615.
[32]	M. R. Kosorok, Introduction to empirical processes and semiparametric inference, Berlin: Springer, 2008. https://doi.org/10.1007/978-0-387-74978-5_15
[33]	P. J. Bickel, C. A. J. Klaassen, Y. Ritov, J. A. Wellner, Efficient and adaptive estimation for semiparametric models, Berlin: Springer, 1993.
[34]	C. Yuan, S. Zhao, S. Li, X. Song, Sieve maximum likelihood estimation of partially linear transformation models with interval-censored data, Stat. Med., 43 (2024), 5765–5776. https://doi.org/10.1002/sim.10225 doi: 10.1002/sim.10225

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)