The Annals of Statistics

Current status linear regression

Piet Groeneboom and Kim Hendrickx

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We construct $\sqrt{n}$-consistent and asymptotically normal estimates for the finite dimensional regression parameter in the current status linear regression model, which do not require any smoothing device and are based on maximum likelihood estimates (MLEs) of the infinite dimensional parameter. We also construct estimates, again only based on these MLEs, which are arbitrarily close to efficient estimates, if the generalized Fisher information is finite. This type of efficiency is also derived under minimal conditions for estimates based on smooth nonmonotone plug-in estimates of the distribution function. Algorithms for computing the estimates and for selecting the bandwidth of the smooth estimates with a bootstrap method are provided. The connection with results in the econometric literature is also pointed out.

Article information

Ann. Statist., Volume 46, Number 4 (2018), 1415-1444.

Received: June 2016
Revised: March 2017
First available in Project Euclid: 27 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62N01: Censored data models
Secondary: 62-04: Explicit machine computation and programs (not the theory of computation or programming)

Current status linear regression MLE semiparametric model


Groeneboom, Piet; Hendrickx, Kim. Current status linear regression. Ann. Statist. 46 (2018), no. 4, 1415--1444. doi:10.1214/17-AOS1589.

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  • [1] Chen, X., Linton, O. and Van Keilegom, I. (2003). Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71 1591–1608.
  • [2] Cosslett, S. R. (1983). Distribution-free maximum likelihood estimator of the binary choice model. Econometrica 51 765–782.
  • [3] Cosslett, S. R. (1987). Efficiency bounds for distribution-free estimators of the binary choice and the censored regression models. Econometrica 55 559–585.
  • [4] Cosslett, S. R. (2007). Efficient estimation of semiparametric models by smoothed maximum likelihood. Internat. Econom. Rev. 48 1245–1272.
  • [5] Ding, Y. and Nan, B. (2011). A sieve $M$-theorem for bundled parameters in semiparametric models, with application to the efficient estimation in a linear model for censored data. Ann. Statist. 39 3032–3061.
  • [6] Dominitz, J. and Sherman, R. P. (2005). Some convergence theory for iterative estimation procedures with an application to semiparametric estimation. Econometric Theory 21 838–863.
  • [7] Finkelstein, D. M. (1986). A proportional hazards model for interval-censored failure time data. Biometrics 42 845–854.
  • [8] Finkelstein, D. M. and Wolfe, R. A. (1985). A semiparametric model for regression analysis of interval-censored failure time data. Biometrics 41 933–945.
  • [9] Giné, E. and Nickl, R. (2015). Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge Univ. Press, New York.
  • [10] Groeneboom, P. and Hendrickx, K. (2018). Supplement to “Current status linear regression.” DOI:10.1214/17-AOS1589SUPP.
  • [11] Groeneboom, P. and Jongbloed, G. (2014). Nonparametric Estimation Under Shape Constraints: Estimators, Algorithms and Asymptotics. Cambridge Series in Statistical and Probabilistic Mathematics 38. Cambridge Univ. Press, New York.
  • [12] Groeneboom, P., Jongbloed, G. and Witte, B. I. (2010). Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model. Ann. Statist. 38 352–387.
  • [13] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. DMV Seminar 19. Birkhäuser, Basel.
  • [14] Han, A. K. (1987). Nonparametric analysis of a generalized regression model. The maximum rank correlation estimator. J. Econometrics 35 303–316.
  • [15] Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models. Ann. Statist. 21 157–178.
  • [16] Höffding, W. (1940). Maszstabinvariante Korrelationstheorie. Schr. Math. Inst. U. Inst. Angew. Math. Univ. Berlin 5 181–233. Translated in: The Collected Works of Wassily Hoeffding (N. I. Fisher and P. K. Sen, eds.). Springer, New York, 1994.
  • [17] Huang, J. (1996). Efficient estimation for the proportional hazards model with interval censoring. Ann. Statist. 24 540–568.
  • [18] Huang, J. and Wellner, J. (1993). Regression models with interval censoring. In Proceedings of the Kolmogorov Seminar. Euler Mathematics Institute, St. Petersburg, Russia.
  • [19] Klein, R. W. and Spady, R. H. (1993). An efficient semiparametric estimator for binary response models. Econometrica 61 387–421.
  • [20] Kosorok, M. R. (2008). Bootstrapping in Grenander estimator. In Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen. Inst. Math. Stat. (IMS) Collect. 1 282–292. IMS, Beachwood, OH.
  • [21] Li, G. and Zhang, C.-H. (1998). Linear regression with interval censored data. Ann. Statist. 26 1306–1327.
  • [22] Murphy, S. A., van der Vaart, A. W. and Wellner, J. A. (1999). Current status regression. Math. Methods Statist. 8 407–425.
  • [23] Rabinowitz, D., Tsiatis, A. and Aragon, J. (1995). Regression with interval-censored data. Biometrika 82 501–513.
  • [24] Rossini, A. J. and Tsiatis, A. A. (1996). A semiparametric proportional odds regression model for the analysis of current status data. J. Amer. Statist. Assoc. 91 713–721.
  • [25] Sen, B., Banerjee, M. and Woodroofe, M. (2010). Inconsistency of bootstrap: The Grenander estimator. Ann. Statist. 38 1953–1977.
  • [26] Sen, B. and Xu, G. (2015). Model based bootstrap methods for interval censored data. Comput. Statist. Data Anal. 81 121–129.
  • [27] Shen, X. (2000). Linear regression with current status data. J. Amer. Statist. Assoc. 95 842–852.
  • [28] Sherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator. Econometrica 61 123–137.
  • [29] Shiboski, S. C. and Jewell, N. P. (1992). Statistical analysis of the time dependence of HIV infectivity based on partner study data. J. Amer. Statist. Assoc. 87 360–372.
  • [30] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.

Supplemental materials

  • Supplement to “Current status linear regression”. We give the proofs of the results stated in Sections 3, 4 and 5 of the manuscript.