NONPARAMETRIC REGRESSION ANALYSIS BASED ON RATIONAL (Padé) APPROXIMATION FOR CENSORED-DATA

Abstract

This paper considers the estimation of a nonparametric regression model with randomly right-censored data. To estimate the model, rational (Padé) approximation based on truncated total least squares (P-TTLS) is used as a smoothing method. Because of censored, data points cannot be used directly in modeling process, a data transformation is needed for overcoming this problem. As known, synthetic data transformation assigns censored points as zero and gives additional magnitudes to uncensored ones associated with Kaplan-Meier distribution of the censored dataset. Thus, the differences between censored and uncensored observations grow which causes a kind of spatial variation in the shape of data. In this paper, to bring a solution to this problematic situation, P-TTLS is used that works well on spatial variation. Also, to see the performance of the P-TTLS on censored data modeling, a simulation study is carried out and it is compared with the benchmarked kernel smoothing (B-KS) method to observe how P-TTLS behaves.

Keywords

• Pade approximation
• right-censored data
• kernel smoothing
• truncated total least squares

References

1. [1] Cox DR. Regression models and life-tables, Journal of Royal Statistical Society, Series B, 1972; 34(2), 187-220
2. [2] Cox DR. Partial likelihood, Biometrika, 1975; 62(2), 269-276.
3. [3] Tsiatis AA. A large sample study of Cox’s regression model, The Annals of Statistics, 1981; 9(1), 93-108.
4. [4] Andersen PK and Gill RD. Cox’s regression model for counting processes: A large sample study, The Annals of Statistics, 1982; 10(4), 1100-1120.
5. [5] Miller RG. Least squares regression with censored data, Biometrika, 1976; 63, 449-64.
6. [6] Stute W and Wang J-L. The strong law under random censorship, The Annals of Statistics, 1993; 21(3), 1591-1607.
7. [7] Stute W. The central limit theorem under random censorship, The Annals of Statistics, 1995; 23(2), 422-439.
8. [8] Stute W. Consistent estimation under random censorship when covariables are present, Journal of Multivariate Analysis, 1993; 45(1), 89-103.

9. [9] Yılmaz E and Aydın D. Bandwidth selection problem for nonparametric regression model with right-censored data, Romanian Statistical Review, 2017; 2, 81-104.
10. [10] Orbe J and Virto J. Penalized spline smoothing using Kaplan-Meier weights with censored data, Biometrical Journal, 2018; 60(5), doi:10.1002/bimj.201700213.
11. [11] Buckley J and James I. Linear regression with censored data, Biometrika, 1979; 66(3), 429-436.
12. [12] Koul H, Susarla V and Van Ryzin J. Regression analysis with randomly right-censored data, The Annals of Statistics, 1981; 9, 1276-1288.
13. [13] Leurgans S. Linear models, random censoring and synthetic data, Biometrika, 1987; 74(2), 301-309.
14. [14] Dabrowska DM. Rank tests for independence for bivariate censored data, The Annals of Statistics, 1986; 14(1), 250, 264.
15. [15] Gross ST and Lai TL. Bootstrap methods for truncated and censored data, Statistica Sinica, 1996; 6, 509-530.
16. [16] Kim HT and Truong YK. Nonparametric regression estimates with censored data: Local linear smoothers and their applications, Biometrics, 1998; 54(4), 1434-1444.
17. [17] Ghouch AE, Van Keilegom I. Nonparametric regression with dependent censored data, Scandinavian Journal of Statistics, 2008; 35(2), 228-247.
18. [18] Osman M and Ghosh SK. Nonparametric regression models for right-censored data using Bernstein polynomials, Computational Statistics & Data Analysis, 2012; 56(3), 559-573.
19. [19] Zhang D and Cherkaev E. Pade approximations for identification of air bubble volume from temperature or frequency dependent permittivity of a two-component mixture, 2008; 16(4), 425-445.
20. [20] Aydın D and Yılmaz E. Truncation level selection in nonparametric regression usig Padé approximation, Communications in Statistics- Simulation and Computation, 2019; doi: 10.1080/03610918.2019.1565586.
21. [21] Aydın D and Yılmaz E. Modified spline regression based on randomly right-censored data: A comparative study, Communications in Statistics - Simulation and Computation, 2018; 47(9), 2587-2611.
22. [22] Fierro RD, Golub GH, Hansen PC and O’Leary DP. Regularization by truncated total least squares. SIAM Journal on Scientific Computing, 1997; 18(4), 1223–41.
23. [23] Sima DM and Huffel S. Level choice in truncated total least squares. Computational Statistics and Data Analysis, 2007; 52, 1208–22.
24. [24] Nadaray, EA. On nonparametric estimates of density functions and regression curves,Theory Appl. Probability, 1964; 10, 186–190,
25. [25] Watson GS. Smooth regression analysis, Sankhya, 1964; 26 (15), 175–184