Abstract
This study considers the nonparametric estimation of a regression function when the response variable is the waiting time between two consecutive events of a stationary renewal process, and where this variable is not completely observed. In these circumstances, our data are the recurrence times from the occurrence of the last event up to a pre-established time, along with the corresponding values of a certain set of covariates. Estimation of the error density function and some of its characteristics are also considered. For the proposed estimators, we first analyze their asymptotic behavior and, thereafter, carry out a simulation study to highlight their behavior in finite samples. Finally, we apply this methodology to an illustrative example with biomedical data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alcalá JT, Cristóbal JA, González-Manteiga W (1999) Goodness-of-fit test for linear models based on local polynomials. Stat Probab Lett 44:39–46
Armstrong PW (2000) First steps in analysing NHS waiting times: avoiding the stationary and closed population’ fallacy. Stat Med 19:2037–2051
Babu GJ, Canty AJ, Chaubey YP (2002) Application of Bernstein Polynomials for smooth estimation of a distribution and density function. J Stat Plann Inference 105:377–392
Barmi HEl, Nelson PI (2002) A note on estimating a non-increasing density in the presence of selection bias. J Stat Plann Inference 107:353–364
Chang S-H (2004) Estimating marginal effects in accelerated failure time models for serial sojourn times among repeated events. Lifetime Data Anal 10:175–190
Chang S-H, Tzeng S-J (2006) Nonparametric estimation of sojourn time distributions for truncated serial event data—a weight-adjusted approach. Lifetime Data Anal 12:53–67
Cox DR (1997) Some remarks on the analysis of survival data. In: Proceedings of the first Seattle symposium in biostatistics: survival analysis. Lecture notes in statistics, vol 123. Springer, pp 1–9
Cristóbal JA, Alcalá JT (2000) Nonparametric regression estimators for length biased data. J Stat Plann Inference 89:145–168
Cristóbal JA, Alcalá JT (2001) An overview of nonparametric contributions to the problem of functional estimation from biased data. Test 10(2):309–332
Cristóbal JA, Ojeda JL, Alcalá JT (2004) Confidence bands in nonparametric regression with length biased data. Ann Inst Stat Math 56:475–496
van Es B, Klaassen CAJ, Oudshoorn K (2000) Survival analysis under cross-sectional sampling: length bias and multiplicative censoring. J Stat Plann Inference 91:295–312
Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman and Hall, London
Hutton JL, Monaghan PF (2002) Choice of parametric accelerated life and proportional hazards models for survival data: asymptotic results. Lifetime Data Anal 8:375–393
Kalbfleisch JD, Prentice RL (1980) The statistical analysis of failure time data. Wiley, New York
Klaassen CAJ, Mokveld PJ, van Es B (2003) Efficient estimation in the accelerated failure time model under cross sectional sampling Technical Report, ArXiv:math.ST/0305234v1
Keiding N, Kvist K, Hartvig H, Tvede M, Juul S (2002) Estimating time to pregnancy from current durations in a cross-sectional sample. Biostatistics 3(4):565–578
Kulikov VN, Lopuhaä HP (2005) Asymptotic normality of the L k -error of the Grenander estimator. Ann Stat 33:2228–2255
Ojeda JL, Cristóbal JA, Alcalá JT (2004) Nonparametric confidence bands construction for GLM models with length biased data. J Nonparametric Stat 16:421–441
Prakasa Rao BLS (1983) Nonparametric function estimation. Wiley, New York
Reid N (1994) A conversation with Sir David Cox. Stat Sci 9:439–455
Siciliani L, Hurst J (2005) Tackling excessive waiting times for elective surgery: a comparative analysis of policies in 12 OECD countries. Health Policy 72:201–215
Sobolev B, Brown P, Zelt D, Kuramoto L (2004) Waiting time in relation to wait-list size at registration: statistical analysis of a waiting-list registry. Clin Invest Med 27:298–305
Solomon PJ (1984) Effect of misspecification of regression models in the analysis of survival data. Biometrika 71:291–198
Struthers CA, Kalbfleisch JD (1986) Misspecified proportional hazards models. Biometrika 73: 363–369
Sun J, Woodroofe M (1996) Adaptive smoothing for a penalized NPMLE of a non-increasing density. J Stat Plann Inference 52:143–159
Vardi Y (1982) Nonparametric estimation in renewal processes. Ann Stat 10:772–785
Vardi Y (1989) Multiplicative censoring, renewal processes, deconvolution and decreasing density: nonparametric estimation. Biometrika 76(4):751–761
Vardi Y, Zhang CH (1992) Large sample study of empirical distributions in a random-multiplicative censoring model. Ann Stat 20(2):1022–1039
Woodroofe M, Sun J (1993) A penalized maximum likelihood estimate of f(0+) when f is non-increasing. Stat Sin 3:501–515
Wu CO (2000) Local Polynomial regression with selection biased data. Stat Sin 10:789–817
Zelen M (2004) Forward and backward recurrence times and length biased sampling; age specific models. Lifetime Data Anal 10:325–334
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cristóbal, J.A., Alcalá, J.T. & Ojeda, J.L. Nonparametric estimation of a regression function from backward recurrence times in a cross-sectional sampling. Lifetime Data Anal 13, 273–293 (2007). https://doi.org/10.1007/s10985-007-9033-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10985-007-9033-5