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Parametric likelihood inference and goodness-of-fit for dependently left-truncated data, a copula-based approach

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  • Takeshi Emura

    (National Central University)

  • Chi-Hung Pan

    (National Central University)

Abstract

Traditionally, the literature on statistical inference with left-truncated samples assumes the independence of truncation variable on lifetime. Alternatively, this paper considers an approach of using a copula for dependent truncation. When considering maximum likelihood estimation and goodness-of-fit procedures, key challenges are the absence of the explicit form of the inclusion probability and truncated distribution functions. This paper shows that, under the copula model, the inclusion probability and truncated distribution functions are expressed as univariate integrals of some functions. With aid of these expressions, we propose computational algorithms to maximize the log-likelihood and to perform goodness-of-fit tests. Simulations are conducted to examine the performance of the proposed method. Real data from a field reliability study on the brake pad lifetimes are analyzed for illustration. Relevant computational programs are made available in the R package “depend.truncation”.

Suggested Citation

  • Takeshi Emura & Chi-Hung Pan, 2020. "Parametric likelihood inference and goodness-of-fit for dependently left-truncated data, a copula-based approach," Statistical Papers, Springer, vol. 61(1), pages 479-501, February.
    • Handle: RePEc:spr:stpapr:v:61:y:2020:i:1:d:10.1007_s00362-017-0947-z
    DOI: 10.1007/s00362-017-0947-z
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    References listed on IDEAS

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    4. Emura, Takeshi & Shiu, Shau-Kai, 2014. "Estimation and model selection for left-truncated and right-censored lifetime data with application to electric power transformers analysis," MPRA Paper 57528, University Library of Munich, Germany.
    5. Joe, H., 1993. "Parametric Families of Multivariate Distributions with Given Margins," Journal of Multivariate Analysis, Elsevier, vol. 46(2), pages 262-282, August.
    6. Emura, Takeshi & Chen, Yi-Hau, 2014. "Gene selection for survival data under dependent censoring: a copula-based approach," MPRA Paper 58043, University Library of Munich, Germany.
    7. Martin, Emily C. & Betensky, Rebecca A., 2005. "Testing Quasi-Independence of Failure and Truncation Times via Conditional Kendall's Tau," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 484-492, June.
    8. Filippo Domma & Sabrina Giordano, 2013. "A copula-based approach to account for dependence in stress-strength models," Statistical Papers, Springer, vol. 54(3), pages 807-826, August.
    9. T. Emura & K. Murotani, 2015. "An algorithm for estimating survival under a copula-based dependent truncation model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 734-751, December.
    10. Lajmi Lakhal Chaieb & Louis-Paul Rivest & Belkacem Abdous, 2006. "Estimating survival under a dependent truncation," Biometrika, Biometrika Trust, vol. 93(3), pages 655-669, September.
    11. Emura, Takeshi & Wang, Weijing, 2012. "Nonparametric maximum likelihood estimation for dependent truncation data based on copulas," Journal of Multivariate Analysis, Elsevier, vol. 110(C), pages 171-188.
    12. Ya-Hsuan Hu & Takeshi Emura, 2015. "Maximum likelihood estimation for a special exponential family under random double-truncation," Computational Statistics, Springer, vol. 30(4), pages 1199-1229, December.
    13. Ulf Schepsmeier & Jakob Stöber, 2014. "Derivatives and Fisher information of bivariate copulas," Statistical Papers, Springer, vol. 55(2), pages 525-542, May.
    14. Emura, Takeshi & Wang, Weijing, 2010. "Testing quasi-independence for truncation data," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 223-239, January.
    15. Luca Greco & Laura Ventura, 2011. "Robust inference for the stress–strength reliability," Statistical Papers, Springer, vol. 52(4), pages 773-788, November.
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    Cited by:

    1. Ning, Jing & Pak, Daewoo & Zhu, Hong & Qin, Jing, 2022. "Conditional independence test of failure and truncation times: Essential tool for method selection," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    2. Kosuke Nakazono & Yu-Cheng Lin & Gen-Yih Liao & Ryuji Uozumi & Takeshi Emura, 2024. "Computation of the Mann–Whitney Effect under Parametric Survival Copula Models," Mathematics, MDPI, vol. 12(10), pages 1-22, May.
    3. Emura, Takeshi & Hsu, Jiun-Huang, 2020. "Estimation of the Mann–Whitney effect in the two-sample problem under dependent censoring," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    4. Nanami Taketomi & Kazuki Yamamoto & Christophe Chesneau & Takeshi Emura, 2022. "Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch," Mathematics, MDPI, vol. 10(20), pages 1-23, October.
    5. Achim Dörre & Chung-Yan Huang & Yi-Kuan Tseng & Takeshi Emura, 2021. "Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model," Computational Statistics, Springer, vol. 36(1), pages 375-408, March.

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