Abstract
We discuss the variance estimation for the nonparametric distribution estimator for doubly censored data. We first provide another view of Kuhn–Tucker’s conditions to construct the profile likelihood, and lead a Newton–Raphson algorithm as an optimization technique unlike the EM algorithm. The main proposal is an iteration-free Wald-type variance estimate based on the chain rule of differentiating conditions to construct the profile likelihood, which generalizes the variance formula in only right- or left-censored data. In this estimation procedure, we overcome some difficulties caused in directly applying Turnbull’s formula to large samples and avoid a load with computationally heavy iterations, such as solving the Fredholm equations, computing the profile likelihood ratio or using the bootstrap. Also, we establish the consistency of the formulated Wald-type variance estimator. In addition, simulation studies are performed to investigate the properties of the Wald-type variance estimates in finite samples in comparison with those from the profile likelihood ratio.
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References
Chang M.N. (1990) Weak convergence of a self-consistent estimator of the survival function with doubly censored data. Annals of Statistics 18: 391–404
Chang M.N., Yang G.L. (1987) Strong consistency of a nonparametric estimator of the survival function with doubly censored data. Annals of Statistics 15: 1536–1547
Chen K., Zhou M. (2003) Non-parametric hypothesis testing and confidence intervals with doubly censored data. Lifetime Data Analysis 9: 71–91
Gill R.D., Johansen S. (1990) A survey of product-integration with a view toward application in survival analysis. Annals of Statistics 18: 1501–1555
Gu M.G., Zhang C.-H. (1993) Asymptotic properties of self-consistent estimators based on doubly censored data. Annals of Statistics 21: 611–624
Klein J.P., Moeschberger M.L. (2003) Survival analysis: techniques for censored and truncated data, (2nd ed.). Springer, New York
Murphy S.A., van der Vaart A.W. (1997) Semiparametric likelihood ratio inference. Annals of Statistics 25: 1471–1509
Mykland P.A., Ren J.-J. (1996) Algorithms for computing self-consistent and maximum likelihood estimators with doubly censored data. Annals of Statistics 24: 1740–1764
Sugimoto T., Hamasaki T. (2006) Properties of estimators of baseline hazard functions in a semiparametric cure model. Annals of the Institute of Statistical Mathematics 58: 647–674
Turnbull B.W. (1974) Nonparametric estimation of a survivorship function with doubly censored data. Journal of the American Statistical Association 69: 169–173
Turnbull B.W. (1976) The empirical distribution function with arbitrarily grouped, censored and truncated data. Journal of the Royal Statistical Society: Series B 38: 290–295
Wellner J.A., Zhan Y. (1997) A hybrid algorithm for computation of the nonparametric maximum likelihood estimator from censored data. Journal of the American Statistical Association 92: 945–959
Zhu C., Sun J. (2007) Variance estimation of a survival function with doubly censored failure time data. In: Auget J.-L., Balakrishnan N., Mesbah M., Molenberghs G. (eds) Advances in statistical methods for the health sciences. Birkhäuser, Boston, pp 225–235
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An erratum to this article can be found at http://dx.doi.org/10.1007/s10463-011-0331-z
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Sugimoto, T. A Wald-type variance estimation for the nonparametric distribution estimators for doubly censored data. Ann Inst Stat Math 63, 645–670 (2011). https://doi.org/10.1007/s10463-009-0251-3
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DOI: https://doi.org/10.1007/s10463-009-0251-3