Electronic Journal of Statistics

Semiparametric additive transformation model under current status data

Guang Cheng and Xiao Wang

Full-text: Open access

Abstract

We consider the efficient estimation of the semiparametric additive transformation model with current status data. A wide range of survival models and econometric models can be incorporated into this general transformation framework. We apply the B-spline approach to simultaneously estimate the linear regression vector, the nondecreasing transformation function, and a set of nonparametric regression functions. We show that the parametric estimate is semiparametric efficient in the presence of multiple nonparametric nuisance functions. An explicit consistent B-spline estimate of the asymptotic variance is also provided. All nonparametric estimates are smooth, and shown to be uniformly consistent and have faster than cubic rate of convergence. Interestingly, we observe the convergence rate interfere phenomenon, i.e., the convergence rates of B-spline estimators are all slowed down to equal the slowest one. The constrained optimization is not required in our implementation. Numerical results are used to illustrate the finite sample performance of the proposed estimators.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 1735-1764.

Dates
First available in Project Euclid: 13 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1323785607

Digital Object Identifier
doi:10.1214/11-EJS656

Mathematical Reviews number (MathSciNet)
MR2870149

Zentralblatt MATH identifier
1329.62197

Subjects
Primary: 60G20: Generalized stochastic processes 62N10
Secondary: 62F12: Asymptotic properties of estimators

Keywords
B-spline consistent variance estimation current status data efficient estimation semiparametric transformation models

Citation

Cheng, Guang; Wang, Xiao. Semiparametric additive transformation model under current status data. Electron. J. Statist. 5 (2011), 1735--1764. doi:10.1214/11-EJS656. https://projecteuclid.org/euclid.ejs/1323785607


Export citation

References

  • [1] Banerjee, M., Biswas, P. and Ghosh, D. (2006). A Semiparametric Binary Regression Model involving Monotonicity Constraints., Scandinavian Journal of Statistics, 33 673–697.
  • [2] Banerjee, M., Mukherjee, D. and Mishra, S. (2009). Semiparametric Binary Regression Models under Shape Constraints with an Application to Indian Schooling Data., Journal of Econometrics. 149 101–117.
  • [3] Bickel, P., Klaassen, C.A., Ritov, Y. and Wellner, J.A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ., Press
  • [4] Chen, K. and Tong, X. (2010). Varying Coefficient Transformation Models with Censored Data., Biometrika, 97 969–976.
  • [5] Chen, X. and Shen, X. (1998). Sieve Extremum Estimates for Weakly Dependent Data., Econometrica, 66 289–314.
  • [6] Cheng, G. and Huang, J. (2010). Bootstrap Consistency for General Semiparametric M-estimation., Annals of Statistics, 38 2884–2915.
  • [7] Dabrowska, D.M. and Doksum, K.A. (1988). Partial Likelihood in Transformation Models with Censored Data., Scandinavian Journal of Statistics, 15 1–23.
  • [8] Dabrowska, D.M. and Doksum, K.A. (1988). Estimation and Testing in a Two-Sample Generalized Odds Rate Model., Journal of American Statistical Association, 83 744–749.
  • [9] Doksum, K.A. and Gasko, M. (1990). On a Correspondence between Models in Binary Regression Analysis and in Survival Analysis., Journal of American Statistical Association, 83 744–749.
  • [10] Huang, J. and Rossini, A.J. (1997). Sieve Estimation for the Proportional-Odds Failure-Time Regression Model with Interval Censoring., Journal of American Statistical Association, 92 960–967.
  • [11] Huang, J. (1999). Efficient Estimation of the Partly Linear Additive Cox Model., Annals of Statistics, 27 1536–1563.
  • [12] Kalbfleisch, J.D. and Prentice, R.L. (1980)., The Statistical Analysis of Failure Time Data. John Wiley and Sons, New York.
  • [13] Kosorok, M.R. (2008)., Introduction to Empirical Processes and Semiparametric Inference. Springer, New York.
  • [14] Lam, K.F. and Xue, H. (2005). A semiparametric regression cure model with current status data., Biometrika, 92 573–586.
  • [15] Ma, S. (2009). Cure model with current status data., Statistica Sinica, 19 233–249.
  • [16] Ma, S. and Kosorok, M.R. (2005a). Penalized Log-likelihood Estimator for Partly Linear Transformation Models with Current Status Data., Annals of Statistics, 33 2256–2290.
  • [17] Ma, S. and Kosorok, M.R. (2005b). Robust Semiparametric M-estimation and the Weighted Bootstrap., Journal of Multivariate Statistics, 96 190–217.
  • [18] Radchenko, P. (2008). Mixed-Rates Asymptotics., Annals of Statistics, 36 287–309.
  • [19] Sasieni, P. (1992). Nan-orthogonal Projections and Their Application to Calculating the Information in a Partly Linear Cox Model., Scandinavian Journal of Statistics 19 215–233.
  • [20] Shen, X. (1998). Proportional Odds Regression and Sieve Maximum Likelihood Estimation., Biometrika, 85 165–177.
  • [21] Shen, X. (2000). Linear Regression with Current Status Data., Journal of American Statistical Association, 95 842–852.
  • [22] Stone, C. (1982). Optimal Global Rates of Convergence for Nonparametric Regression., Annals of Statistics, 10 1040–1053.
  • [23] Stone, C. (1985). Additive Regression and Other Nonparametric Models., Annals of Statistics, 13 689–705.
  • [24] Van de Geer, S. (2000). Empirical Processes in M-Estimation. Cambridge University, Press.
  • [25] van der Vaart, A. W., and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New, York
  • [26] Xue, H., Lam, K.F., and Li, G. (2004). Sieve maximum likelihood estimator for semiparametric regression models with current status data., Journal of the American Statistical Association, 99 346–356.
  • [27] Yu, A.K.F., Kwan, K.Y.W., Chan, D.H.Y., and Fong, D.Y.T. (2001). Clinical features of 46 eyes with calcified hydrogel intraocular lenses., Journal of Cataract and Refractive Surgery, 27 1596–1606.
  • [28] Zhang, Y., Hua, L. and Huang, J. (2010). A Spline-based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval-Censored Data, Scandinavian Journal of Statistics 37 338–354.