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Generalized Cramér-von Mises tests of goodness of fit for doubly censored data

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Abstract

We generalize Cramér-von Mises statistics to test the goodness of fit of a lifetime distribution when the data are doubly censored. We derive the limiting distributions of our test statistics under the null hypothesis and the alternative hypothesis, respectively. We also give a strong consistent estimator for the asymptotic covariance of the self-consistent estimator for the survival function with doubly censored data. Thereby, a method, called the Fredholm Integral Equation method, is proposed to estimate the null distribution of test statistics. In this work, the perturbation theory for linear operators plays an important role, and some numerical examples are included.

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References

  • Bickel, P. J. and Ren, J. (1995). Them out ofn bootstrap and goodness of fit tests with doubly censored data (preprint).

  • Burden, R. L. and Faires, J. D. (1989).Numerical Analysis, PWS-KENT Publishing Company, Boston, Massachusetts.

    Google Scholar 

  • Chang, M. N. (1990). Weak convergence of a self-consistent estimator of the survival function with doubly censored data,Ann. Statist.,18, 391–404.

    Google Scholar 

  • Chang, M. N. and Yang, G. L. (1987). Strong consistency of a nonparametric estimator of the survival function with doubly censored data,Ann. Statist.,15, 1536–1547.

    Google Scholar 

  • Delves, L. M. and Mohamed, J. L. (1985).Computational Methods for Integral Equations, Cambridge University Press, Massachusetts.

    Google Scholar 

  • Delves, L. M. and Walsh, J. (1974).Numerical Solution of Integral Equations, Clarendon Press, Oxford.

    Google Scholar 

  • Fernholz, L. T. (1983). Von Mises calculus for statistical functional,Lecture Notes in Statist.,19, Springer, New York.

    Google Scholar 

  • Gehan, E. A. (1965). A generalized two-sample Wilcoxon test for doubly censored data,Biometrika,52, 650–653.

    Google Scholar 

  • Gill, R. D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method (part I),Scand. J. Statist.,16, 97–128.

    Google Scholar 

  • Gu, M. G. and Zhang, C. H. (1993). Asymptotic properties of self-consistent estimators based on doubly censored data,Ann. Statist.,21(2), 611–624.

    Google Scholar 

  • Kato, T. (1980).Perturbation Theory for Linear Operators, Springer, New York.

    Google Scholar 

  • Loève, M. (1963).Probability Theory, Van Nostrand, Princeton, New Jersey.

    Google Scholar 

  • Mantel, N. (1967). Ranking procedures for arbitrarily restricted observations,Biometrics,23, 65–78.

    Google Scholar 

  • Marron, J. S. and Padgett, W. J. (1987). Asymptotically optimal bandwidth selection for kernel density estimators from randomly right-censored samples,Ann. Statist.,15(4), 1520–1535.

    Google Scholar 

  • Mielniczuk, J. (1986). Some asymptotic properties of kernel estimators of a density function in case of censored data,Ann. Statist.,14(2), 766–773.

    Google Scholar 

  • Peto, R. (1973). Experimental survival curves for interval-censored data,Appl. Statist.,22, 86–91.

    Google Scholar 

  • Ren, J. (1994). On self-consistent estimators and kernel density estimators with doubly censored data, Tech. Report, #22, Division of Statistics, Department of Mathematics & Statistics, University of Nebraska-Lincoln (submitted).

  • Ren, J. and Ledder, G. (1995). On Cramér-von Mises goodness of fit tests for doubly censored data (submitted).

  • Ren, J. and Sen P. K. (1991). On Hadamard differentiability of extended statistical functional,J. Multivariate Anal.,39, 30–43.

    Google Scholar 

  • Ren, J. and Zhou, M. (1993). Extended L-estimators and M-estimators for doubly censored data, Tech. Report, #20, Division of Statistics, Department of Mathematics & Statistics, University of Nebraska-Lincoln (submitted).

  • Ren, J. and Zhou, M. (1994). Generalized R-statistics for doubly censored data, Tech. Report, #25, Division of Statistics, Department of Mathematics & Statistics, University of Nebraska-Lincoln.

  • Rosenblatt, M. (1952). Limit theorems associated with variants of the von Mises statistic,Ann. Math. Statist.,23, 617–623.

    Google Scholar 

  • Tsai, W. Y. and Crowley, J. (1985). A large sample study of generalized maximum likelihood estimators from incomplete data via self-consistency,Ann. Statist.,13, 1317–1334.

    Google Scholar 

  • Turnbull, B. W. (1974). Nonparametric estimation of a survivorship function with doubly censored data,J. Amer. Statist. Assoc.,69, 169–173.

    Google Scholar 

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Authors and Affiliations

  1. Division of Statistics, Department of Mathematics and Statistics, University of Nebraska-Lincoln, 810 Oldfather Hall, P.O. Box 880323, 86588-0323, Lincoln, NE, USA

    Jian-Jian Ren

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  1. Jian-Jian Ren
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The author's research was supported by a Faculty Fellowship of University of Nebraska-Lincoln.

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Ren, JJ. Generalized Cramér-von Mises tests of goodness of fit for doubly censored data. Ann Inst Stat Math 47, 525–549 (1995). https://doi.org/10.1007/BF00773400

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  • DOI: https://doi.org/10.1007/BF00773400

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