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Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses

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  • SOMNATH DATTA
  • DIPANKAR BANDYOPADHYAY
  • GLEN A. SATTEN

Abstract

. A right‐censored version of a U‐statistic with a kernel of degree m 1 is introduced by the principle of a mean preserving reweighting scheme which is also applicable when the dependence between failure times and the censoring variable is explainable through observable covariates. Its asymptotic normality and an expression of its standard error are obtained through a martingale argument. We study the performances of our U‐statistic by simulation and compare them with theoretical results. A doubly robust version of this reweighted U‐statistic is also introduced to gain efficiency under correct models while preserving consistency in the face of model mis‐specifications. Using a Kendall's kernel, we obtain a test statistic for testing homogeneity of failure times for multiple failure causes in a multiple decrement model. The performance of the proposed test is studied through simulations. Its usefulness is also illustrated by applying it to a real data set on graft‐versus‐host‐disease.

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  • Somnath Datta & Dipankar Bandyopadhyay & Glen A. Satten, 2010. "Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 680-700, December.
    • Handle: RePEc:bla:scjsta:v:37:y:2010:i:4:p:680-700
    DOI: 10.1111/j.1467-9469.2010.00697.x
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    1. Isha Dewan & J. V. Deshpande & S. B. Kulathinal, 2004. "On Testing Dependence between Time to Failure and Cause of Failure via Conditional Probabilities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(1), pages 79-91, March.
    2. Somnath Datta & Glen A. Satten, 2002. "Estimation of Integrated Transition Hazards and Stage Occupation Probabilities for Non-Markov Systems Under Dependent Censoring," Biometrics, The International Biometric Society, vol. 58(4), pages 792-802, December.
    3. Bose, Arup & Sen, Arusharka, 2002. "Asymptotic Distribution of the Kaplan-Meier U-Statistics," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 84-123, October.
    4. Ying, Zhiliang, 1989. "A note on the asymptotic properties of the product-limit estimator on the whole line," Statistics & Probability Letters, Elsevier, vol. 7(4), pages 311-314, February.
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    1. Chen, Yichen & Datta, Somnath, 2019. "Adjustments of multi-sample U-statistics to right censored data and confounding covariates," Computational Statistics & Data Analysis, Elsevier, vol. 135(C), pages 1-14.
    2. Dominic Edelmann & Thomas Welchowski & Axel Benner, 2022. "A consistent version of distance covariance for right‐censored survival data and its application in hypothesis testing," Biometrics, The International Biometric Society, vol. 78(3), pages 867-879, September.
    3. Kattumannil, Sudheesh K. & E.P., Sreedevi, 2022. "Non-parametric estimation of cumulative (residual) extropy," Statistics & Probability Letters, Elsevier, vol. 185(C).
    4. Austin, Matthew D. & Betensky, Rebecca A., 2014. "Eliminating bias due to censoring in Kendall’s tau estimators for quasi-independence of truncation and failure," Computational Statistics & Data Analysis, Elsevier, vol. 73(C), pages 16-26.
    5. Marija Cuparić & Bojana Milošević, 2022. "New characterization-based exponentiality tests for randomly censored data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 461-487, June.
    6. Jie Fan & Somnath Datta, 2013. "On Mann–Whitney tests for comparing sojourn time distributions when the transition times are right censored," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 149-166, February.
    7. Sudheesh K. Kattumannil & P. Anisha, 2019. "A simple non-parametric test for decreasing mean time to failure," Statistical Papers, Springer, vol. 60(1), pages 73-87, February.
    8. Kattumannil, Sudheesh K. & Dewan, Isha & N., Sreelaksmi, 2021. "Non-parametric estimation of Gini index with right censored observations," Statistics & Probability Letters, Elsevier, vol. 175(C).
    9. Salim Bouzebda & Thouria El-hadjali & Anouar Abdeldjaoued Ferfache, 2023. "Uniform in Bandwidth Consistency of Conditional U-statistics Adaptive to Intrinsic Dimension in Presence of Censored Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1548-1606, August.
    10. Ao Yuan & Mihai Giurcanu & George Luta & Ming T. Tan, 2017. "U-statistics with conditional kernels for incomplete data models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 271-302, April.
    11. Renjith Mohan & Sreelakshmi N & Sudheesh K. Kattumannil, 2022. "Non-parametric test for decreasing renewal dichotomous Markov noise shock model," Statistical Papers, Springer, vol. 63(3), pages 965-982, June.
    12. Simos G. Meintanis & James Allison & Leonard Santana, 2016. "Goodness-of-fit tests for semiparametric and parametric hypotheses based on the probability weighted empirical characteristic function," Statistical Papers, Springer, vol. 57(4), pages 957-976, December.
    13. Salim Bouzebda & Amel Nezzal & Tarek Zari, 2022. "Uniform Consistency for Functional Conditional U -Statistics Using Delta-Sequences," Mathematics, MDPI, vol. 11(1), pages 1-39, December.

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