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Density and hazard rate estimation for right‐censored data by using wavelet methods

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  • A. Antoniadis
  • G. Grégoire
  • G. Nason

Abstract

This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right‐censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators have pointwise and global mean‐square consistency, obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show that these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its flexibility, provides a new and interesting interpretation.

Suggested Citation

  • A. Antoniadis & G. Grégoire & G. Nason, 1999. "Density and hazard rate estimation for right‐censored data by using wavelet methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(1), pages 63-84.
    • Handle: RePEc:bla:jorssb:v:61:y:1999:i:1:p:63-84
    DOI: 10.1111/1467-9868.00163
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    Cited by:

    1. Bezandry, Paul H. & Bonney, George E. & Gannoun, Ali, 2005. "Consistent estimation of the density and hazard rate functions for censored data via the wavelet method," Statistics & Probability Letters, Elsevier, vol. 74(4), pages 366-372, October.
    2. Elodie Brunel & Fabienne Comte & Agathe Guilloux, 2008. "Estimation Strategies for Censored Lifetimes with a Lexis‐Diagram Type Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 557-576, September.
    3. Li, Linyuan, 2008. "On the block thresholding wavelet estimators with censored data," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1518-1543, September.
    4. Sam Efromovich & Jufen Chu, 2018. "Hazard rate estimation for left truncated and right censored data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 889-917, August.
    5. Liang Han-Ying & Mammitzsch Volker & Steinebach Josef, 2005. "Nonlinear wavelet density and hazard rate estimation for censored data under dependent observations," Statistics & Risk Modeling, De Gruyter, vol. 23(3/2005), pages 161-180, March.
    6. Fabienne Comte & Gwennaelle Mabon & Adeline Samson, 2017. "Spline regression for hazard rate estimation when data are censored and measured with error," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 71(2), pages 115-140, May.
    7. Sam Efromovich, 2016. "Minimax theory of nonparametric hazard rate estimation: efficiency and adaptation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 25-75, February.
    8. Angers, Jean-Francois & MacGibbon, Brenda, 2013. "Hazard function estimation with nonnegative “wavelets”," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 969-978.
    9. Li, Jiexiang & Tran, Lanh Tat, 2007. "Hazard rate estimation on random fields," Journal of Multivariate Analysis, Elsevier, vol. 98(7), pages 1337-1355, August.
    10. Sandra Plancade, 2011. "Model selection for hazard rate estimation in presence of censoring," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 313-347, November.
    11. Fabienne Comte & Adeline Samson & Julien J. Stirnemann, 2018. "Hazard estimation with censoring and measurement error: application to length of pregnancy," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(2), pages 338-359, June.
    12. E. Brunel & F. Comte & A. Guilloux, 2009. "Nonparametric density estimation in presence of bias and censoring," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(1), pages 166-194, May.
    13. Rigat, F. & Mira, A., 2012. "Parallel hierarchical sampling: A general-purpose interacting Markov chains Monte Carlo algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1450-1467.
    14. Liang, Han-Ying & de Uña-Álvarez, Jacobo, 2011. "Wavelet estimation of conditional density with truncated, censored and dependent data," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 448-467, March.

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