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Integrated Square Error of Hazard Rate Estimation for Survival Data with Missing Censoring Indicators

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Abstract

The problem of hazard rate estimation under right-censored assumption has been investigated extensively. Integrated square error (ISE) of estimation is one of the most widely accepted measurements of the global performance for nonparametric kernel estimation. But there are no results available for ISE of hazard rate estimation under right-censored model with censoring indicators missing at random (MAR) so far. This paper constructs an imputation estimator of the hazard rate function and establish asymptotic normality of the ISE for the kernel hazard rate estimator with censoring indicators MAR. At the same time, an asymptotic representation of the mean integrated square error (MISE) is also presented. The finite sample behavior of the estimator is investigated via one simple simulation.

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References

  1. Upadhyay S, Hazard Rate Function, Wiley Encyclopedia of Operations Research and Management Science, John Wiley & Sons, Inc., New York, 2010.

    Google Scholar 

  2. Diallo A and Louani D, Moderate and large deviation principles for the hazard rate function kernel estimator under censoring, Statistics and Probability Letters, 2013, 83(3): 735–743.

    Article  MathSciNet  Google Scholar 

  3. Sun L and Zheng Z, The asymptotic of the integrated square error for the kernel hazard rate estimators with left truncated and right censored data, Systems Science and Mathematical Sciences, 1999, 12(3): 251–262.

    MathSciNet  MATH  Google Scholar 

  4. Cai Z, Kernel density and hazard rate estimation for censored dependent data, Journal of Multivariate Analysis, 1998, 67: 23–34.

    Article  MathSciNet  Google Scholar 

  5. Jomhoori S, Fakoor V, and Azarnoosh H, Central limit theorem for ISE of kernel density estimators in censored dependent model, Communication in Statistics: Theory and Method, 2012, 41: 1334–1349.

    Article  MathSciNet  Google Scholar 

  6. van der Laan M and McKeague I, Efficient estimation from right-censored data when failure indicators are missing at random, The Annals of Statistics, 1998, 26: 164–182.

    Article  MathSciNet  Google Scholar 

  7. McKeague I and Subramanian S, Product-limit estimators and Cox regression with missing censoring information, Scandinavian Journal of Statistics, 1998, 25: 589–601.

    Article  MathSciNet  Google Scholar 

  8. Little R and Rubin D, Statistical Analysis with Missing Data, John Wiley & Sons, New York, 1987.

    MATH  Google Scholar 

  9. Wang Q, Liu W, and Liu C, Probability density estimation for survival data with censoring indicators missing at random, Journal of Multivariate Analysis, 2009, 100: 835–850.

    Article  MathSciNet  Google Scholar 

  10. Li X and Wang Q, The weighted least square based estimators with censoring indicators missing at random, Journal of Statistical Planning and Inference, 2012, 142: 2913–2925.

    Article  MathSciNet  Google Scholar 

  11. Zou Y and Liang H, Wavelet estimation of density for censored data with censoring indicator missing at random, Statistics, 2017, 51(6): 1214–1237.

    Article  MathSciNet  Google Scholar 

  12. Zou Y, Fan G, and Zhang R, Quantile regression and variable selection for partially linear single-index models with missing censoring indicators, Journal of Statistical Planning and Inference, 2020, 204: 80–95.

    Article  MathSciNet  Google Scholar 

  13. Wang Q, Dinse G, and Liu C, Hazard function estimation with cause-of-death data missing at random, The Annals of Statistics, 2012, 64: 415–438.

    MathSciNet  MATH  Google Scholar 

  14. Brunel E, Comte F, and Guilloux A, Nonparametric estimation for survival data with censoring indicators missing at random, Journal of Statistical Planning and Inference, 2013, 143: 1653–1671.

    Article  MathSciNet  Google Scholar 

  15. Wang Q and Ng K, Asymptotically efficient product-limit estimators with censoring indicators missing at random, Statistica Sinica, 2008, 18: 749–768.

    MathSciNet  MATH  Google Scholar 

  16. Serfling R, On the strong law of large numbers and related results for quasi-stationary sequences, Theory of Probability and Its Applications, 1980, 25(1): 187–191.

    Article  Google Scholar 

  17. Hall P, Central limit theorem for integrated square error of multivariate nonparametric density estimators, Journal of Multivariate Analysis, 1984, 14: 1–16.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Economics and Management, Shanghai Maritime University, Shanghai, 201306, China

    Yuye Zou & Guoliang Fan

  2. Key Laboratory of Advanced Theory and Application in Statistics and Data Science - MOE, School of Statistics, East China Normal University, Shanghai, 200062, China

    Riquan Zhang

Authors
  1. Yuye Zou
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  2. Guoliang Fan
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  3. Riquan Zhang
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Corresponding author

Correspondence to Guoliang Fan.

Additional information

This research was supported by the China Postdoctoral Science Foundation under Grant No. 2019M651422, the National Natural Science Foundation of China under Grant Nos. 71701127, 11831008 and 11971171, the National Social Science Foundation Key Program under Grant No. 17ZDA091, the 111 Project of China under Grant No. B14019, the Natural Science Foundation of Shanghai under Grant Nos. 17ZR1409000 and 20ZR1423000 and the Project of Humanities and Social Science Foundation of Ministry of Education under Grant No. 20YJC910003.

This paper was recommended for publication by Editor LI Qizhai.

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Zou, Y., Fan, G. & Zhang, R. Integrated Square Error of Hazard Rate Estimation for Survival Data with Missing Censoring Indicators. J Syst Sci Complex 34, 735–758 (2021). https://doi.org/10.1007/s11424-021-9307-0

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  • DOI: https://doi.org/10.1007/s11424-021-9307-0

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