The Annals of Statistics

Partially monotone tensor spline estimation of the joint distribution function with bivariate current status data

Yuan Wu and Ying Zhang

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The analysis of the joint cumulative distribution function (CDF) with bivariate event time data is a challenging problem both theoretically and numerically. This paper develops a tensor spline-based sieve maximum likelihood estimation method to estimate the joint CDF with bivariate current status data. The $I$-splines are used to approximate the joint CDF in order to simplify the numerical computation of a constrained maximum likelihood estimation problem. The generalized gradient projection algorithm is used to compute the constrained optimization problem. Based on the properties of $B$-spline basis functions it is shown that the proposed tensor spline-based nonparametric sieve maximum likelihood estimator is consistent with a rate of convergence potentially better than $n^{1/3}$ under some mild regularity conditions. The simulation studies with moderate sample sizes are carried out to demonstrate that the finite sample performance of the proposed estimator is generally satisfactory.

Article information

Ann. Statist., Volume 40, Number 3 (2012), 1609-1636.

First available in Project Euclid: 5 September 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Secondary: 60G05: Foundations of stochastic processes

Bivariate current status data constrained maximum likelihood estimation empirical process sieve maximum likelihood estimation tensor spline basis functions


Wu, Yuan; Zhang, Ying. Partially monotone tensor spline estimation of the joint distribution function with bivariate current status data. Ann. Statist. 40 (2012), no. 3, 1609--1636. doi:10.1214/12-AOS1016.

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  • Betensky, R. A. and Finkelstein, D. M. (1999). A non-parametric maximum likelihood estimator for bivariate interval censored data. Stat. Med. 18 3089–3100.
  • Curry, H. B. and Schoenberg, I. J. (1966). On Pólya frequency functions. IV. The fundamental spline functions and their limits. J. Anal. Math. 17 71–107.
  • Dabrowska, D. M. (1988). Kaplan–Meier estimate on the plane. Ann. Statist. 16 1475–1489.
  • de Boor, C. (2001). A Practical Guide to Splines, revised ed. Applied Mathematical Sciences 27. Springer, New York.
  • Diamond, I. D., McDonald, J. W. and Shah, I. H. (1986). Proportional hazards models for current status data: Application to the study of differentials in age at weaning in Pakistan. Demography 23 607–620.
  • Diamond, I. D. and McDonald, J. W. (1991). Analysis of current status data. In Demographic Applications of Event History Analysis (J. Trussell, R. Hankinson and J. Tilton, eds.) 231–252. Oxford Univ. Press, Oxford.
  • Duffy, D. L., Martin, N. G. and Matthews, J. D. (1990). Appendectomy in Australian twins. Am. J. Hum. Genet. 47 590–592.
  • Finkelstein, D. M. and Wolfe, R. A. (1985). A semiparametric model for regression analysis of interval-censored failure time data. Biometrics 41 933–945.
  • Gentleman, R. and Vandal, A. C. (2001). Computational algorithms for censored-data problems using intersection graphs. J. Comput. Graph. Statist. 10 403–421.
  • Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. DMV Seminar 19. Birkhäuser, Basel.
  • Grummer-Strawn, L. M. (1993). Regression analysis of current status data: An application to breat feeding. J. Amer. Statist. Assoc. 88 758–765.
  • Hoel, D. G. and Walburg, H. E. Jr. (1972). Statistical analysis of survival experiments. J. Natl. Cancer Inst. 49 361–372.
  • Huang, J. (1996). Efficient estimation for the proportional hazards model with interval censoring. Ann. Statist. 24 540–568.
  • Huang, J. and Wellner, J. A. (1995). Asymptotic normality of the NPMLE of linear functionals for interval censored data, case 1. Stat. Neerl. 49 153–163.
  • Jamshidian, M. (2004). On algorithms for restricted maximum likelihood estimation. Comput. Statist. Data Anal. 45 137–157.
  • Jewell, N. P., Malani, H. and Vittinghoff, E. (1994). Nonparametric estimation for a form of doubly censored data with application to two problems in AIDS. J. Amer. Statist. Assoc. 89 7–18.
  • Jewell, N. P., van der Laan, M. and Lei, X. (2005). Bivariate current status data with univariate monitoring times. Biometrika 92 847–862.
  • Koo, J. Y. (1996). Bivariate B-splines for tensor logspline density estimation. Comput. Statist. Data Anal. 21 31–42.
  • Kooperberg, C. (1998). Bivariate density estimation with an application to survival analysis. J. Comput. Graph. Statist. 7 322–341.
  • Lu, M. (2010). Spline-based sieve maximum likelihood estimation in the partly linear model under monotonicity constraints. J. Multivariate Anal. 101 2528–2542.
  • Lu, M., Zhang, Y. and Huang, J. (2007). Estimation of the mean function with panel count data using monotone polynomial splines. Biometrika 94 705–718.
  • Lu, M., Zhang, Y. and Huang, J. (2009). Semiparametric estimation methods for panel count data using monotone $B$-splines. J. Amer. Statist. Assoc. 104 1060–1070.
  • Maathuis, M. H. (2005). Reduction algorithm for the NPMLE for the distribution function of bivariate interval-censored data. J. Comput. Graph. Statist. 14 352–362.
  • Meyer, M. C. (2008). Inference using shape-restricted regression splines. Ann. Appl. Stat. 2 1013–1033.
  • Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed. Springer, New York.
  • Prentice, R. L. and Cai, J. (1992). Covariance and survivor function estimation using censored multivariate failure time data. Biometrika 79 495–512.
  • Pruitt, R. C. (1991). Strong consistency of self-consistent estimators: General theory and an application to bivariate survival analysis. Technical report, Dept. Statistics, Univ. Minnesota.
  • Quale, C. M., van der Laan, M. J. and Robins, J. R. (2006). Locally efficient estimation with bivariate right-censored data. J. Amer. Statist. Assoc. 101 1076–1084.
  • Ramsay, J. O. (1988). Monotone regression splines in action. Statist. Sci. 3 425–441.
  • Rosen, J. B. (1960). The gradient projection method for nonlinear programming. I. Linear constraints. J. Soc. Indust. Appl. Math. 8 181–217.
  • Schumaker, L. L. (1981). Spline Functions: Basic Theory. Wiley, New York.
  • Scott, D. W. (1992). Multivariate Density Estimation. Wiley, New York.
  • Shen, X. (1998). Proportional odds regression and sieve maximum likelihood estimation. Biometrika 85 165–177.
  • 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.
  • Shih, J. H. and Louis, T. A. (1995). Inferences on the association parameter in copula models for bivariate survival data. Biometrics 51 1384–1399.
  • Song, S. (2001). Estimation with Bivariate Interval-Censored Data. ProQuest LLC, Ann Arbor, MI.
  • Stone, C. J. (1994). The use of polynomial splines and their tensor products in multivariate function estimation. Ann. Statist. 22 118–184.
  • Sun, L., Wang, L. and Sun, J. (2006). Estimation of the association for bivariate interval-censored failure time data. Scand. J. Stat. 33 637–649.
  • van der Laan, M. J. (1996). Efficient estimation in the bivariate censoring model and repairing NPMLE. Ann. Statist. 24 596–627.
  • van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • Wang, W. and Ding, A. A. (2000). On assessing the association for bivariate current status data. Biometrika 87 879–893.
  • Wang, X. and Shen, J. (2010). A class of grouped Brunk estimators and penalized spline estimators for monotone regression. Biometrika 97 585–601.
  • Wang, Y. and Taylor, J. (2004). Monotone constrained tensor-product B-spline with applicaiton to screen studies. Technical report, Dept. Biostatistics, Univ. Michigan.
  • Wong, G. Y. C. and Yu, Q. (1999). Generalized MLE of a joint distribution function with multivariate interval-censored data. J. Multivariate Anal. 69 155–166.
  • Wu, Y. and Gao, X. (2011). Sieve estimation with bivariate interval censored data. Journal of Statistics: Advances in Theory and Applications 16 37–61.
  • Wu, Y. and Zhang, Y. (2012) Supplement to “Partially monotone tensor spline estimation of the joint distribution function with bivariate current status data.” DOI:10.1214/12-AOS1016SUPP.
  • Zhang, Y., Hua, L. and Huang, J. (2010). A spline-based semiparametric maximum likelihood estimation method for the Cox model with interval-censored data. Scand. J. Stat. 37 338–354.
  • Zhang, S., Zhang, Y., Chaloner, K. and Stapleton, J. T. (2010). A copula model for bivariate hybrid censored survival data with application to the MACS study. Lifetime Data Anal. 16 231–249.

Supplemental materials

  • Supplementary material: Technical lemmas. This supplemental material contains some technical lemmas including their proofs that are imperative for the proofs of Theorems 3.1 and 3.2.