Skip to main content
SpringerLink
Log in

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

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

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”.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Bakoyannis G, Touloumi G (2012) Practical methods for competing risks data: a review. Stat Method Med Res 21:257–272

    MathSciNet  MATH  Google Scholar 

  • Bakoyannis G, Touloumi G (2017) Impact of dependent left truncation in semiparametric competing risks methods: a simulation study. Commun Stat Simul Comput 46(3):2025–2042

    MathSciNet  MATH  Google Scholar 

  • Braekers R, Veraverbeke N (2005) A copula-graphic estimator for the conditional survival function under dependent censoring. Can J Stat 33:429–447

    MathSciNet  MATH  Google Scholar 

  • Chaieb LL, Rivest LP, Abdous B (2006) Estimating survival under a dependent truncation. Biometrika 93:665–669

    MathSciNet  MATH  Google Scholar 

  • Chen YH (2010) Semiparametric marginal regression analysis for dependent competing risks under an assumed copula. J R Stat Soc B 72:235–251

    MathSciNet  MATH  Google Scholar 

  • Cortese G, Ventura L (2013) Accurate higher-order likelihood inference on P(Y \(<\) X). Comput Stat 28(3):1035–1059

    MathSciNet  MATH  Google Scholar 

  • De Uña-Álvarez J (2012) On the Markov three-state progressive model. In: Recent advances in system reliability. Springer, New York

  • Ding AA (2012) Copula identifiability conditions for dependent truncated data model. Lifetime Data Anal 18(4):397–407

    MathSciNet  MATH  Google Scholar 

  • Domma F, Giordano S (2013) A copula-based approach to account for dependence in stress-strength models. Stat Pap 54(3):807–826

    MathSciNet  MATH  Google Scholar 

  • Emura T (2017) R depend.truncation: statistical inference for parametric and semiparametric models based on dependently truncated data, CRAN: version 2.8. https://CRAN.R-project.org/package=depend.truncation

  • Emura T, Chen YH (2016) Gene selection for survival data under dependent censoring: a copula-based approach. Stat Method Med Res 25(6):2840–2857

    MathSciNet  Google Scholar 

  • Emura T, Konno Y (2012a) Multivariate normal distribution approaches for dependently truncated data. Stat Pap 53:133–149

    MathSciNet  MATH  Google Scholar 

  • Emura T, Konno Y (2012b) A goodness-of-fit tests for parametric models based on dependently truncated data. Comput Stat Data Anal 56:2237–2250

    MathSciNet  MATH  Google Scholar 

  • Emura T, Michimae H (2017) A copula-based inference to piecewise exponential models under dependent censoring, with application to time to metamorphosis of salamander larvae. Environ Ecol Stat 24(1):151–173

    MathSciNet  Google Scholar 

  • Emura T, Murotani K (2015) An algorithm for estimating survival under a copula-based dependent truncation model. TEST 24(4):734–751

    MathSciNet  MATH  Google Scholar 

  • Emura T, Shiu SK (2016) Estimation and model selection for left-truncated and right-censored lifetime data with application to electric power transformers analysis. Commun Stat Simul Comput 45(9):3171–3189

    MathSciNet  MATH  Google Scholar 

  • Emura T, Wang W (2010) Testing quasi-independence for truncation data. J Multivar Anal 101:223–239

    MathSciNet  MATH  Google Scholar 

  • Emura T, Wang W (2012) Nonparametric maximum likelihood estimation for dependent truncation data based on copulas. J Multivar Anal 110:171–188

    MathSciNet  MATH  Google Scholar 

  • Emura T, Nakatochi M, Murotani K, Rondeau V (2015) A joint frailty-copula model between tumour progression and death for meta-analysis. Stat Methods Med Res. doi:10.1177/0962280215604510

    MathSciNet  Google Scholar 

  • Emura T, Nakatochi M, Matsui S, Michimae H, Rondeau V (2017a) Personalized dynamic prediction of death according to tumour progression and high-dimensional genetic factors: meta-analysis with a joint model. Stat Methods Med Res. doi:10.1177/0962280216688032

    MathSciNet  Google Scholar 

  • Emura T, Chen YH, Matsui S, Rondeau V (2017b) Survival analysis with dependent censoring and correlated endpoints—copula-based approaches. JSS research series in statistics. Springer, Berlin

  • Emura T, Wang W (2016) Semiparametric inference for an accelerated failure time model with dependent truncation. Ann Inst Stat Math 68(5):1073–1094

    MathSciNet  MATH  Google Scholar 

  • Escarela G, Carriere JF (2003) Fitting competing risks with an assumed copula. Stat Methods Med Res 12(4):333–349

    MathSciNet  MATH  Google Scholar 

  • Gardes L, Stupfler G (2014) Estimating extreme quantiles under random truncation. TEST 24(2):207–227

    MathSciNet  MATH  Google Scholar 

  • Greco L, Ventura L (2011) Robust inference for the stress-strength reliability. Stat Pap 52(4):773–788

    MathSciNet  MATH  Google Scholar 

  • Hong Y, Meeker WQ, Escobar LA (2008) Avoiding problems with normal approximation confidence intervals for probabilities. Technometrics 50(1):64–68

    MathSciNet  Google Scholar 

  • Hong Y, Meeker WQ, McCalley JD (2009) Prediction of remaining life of power transformers based on left truncated and right censored lifetime data. Ann Appl Stat. doi:10.1214/00-AOAS231

    MathSciNet  MATH  Google Scholar 

  • Hsu TM, Emura T, Fan TH (2016) Reliability inference for a copula-based series system life test under multiple type-I censoring. IEEE Trans Reliab 65(2):1069–1080

    Google Scholar 

  • Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation. Comput Stat 30(4):1199–1229

    MathSciNet  MATH  Google Scholar 

  • Joe H (1993) Parametric families of multivariate distributions with given margins. J Multivar Anal 46:262–282

    MathSciNet  MATH  Google Scholar 

  • Kalbfleisch JD, Lawless JF (1992) Some useful statistical methods for truncated data. J Qual Technol 24:145–152

    Google Scholar 

  • Khuri AI (2003) Advanced calculus with applications in statistics (vol 486). Wiley. doi:10.1002/0471394882

  • Klein JP, Moeschberger ML (2003) Survival analysis techniques for censored and truncated data, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Knight K (2000) Mathematical statistics. Chapman and Hall, Boca Raton

    MATH  Google Scholar 

  • Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, Hoboken

    MATH  Google Scholar 

  • Lynden-Bell D (1971) A method of allowing for known observational selection in small samples applied to 3RC quasars. Mon Not R Astron Soc 155:95–118

    Google Scholar 

  • Martin EC, Betensky RA (2005) Testing quasi-independence of failure and truncation via conditional Kendall’s tau. J Am Stat Assoc 100:484–492

    MathSciNet  MATH  Google Scholar 

  • Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Noughabi MS, Kayid M (2017) Bivariate quantile residual life: a characterization theorem and statistical properties. Stat Pap. doi:10.1007/s00362-017-0905-9

    MathSciNet  MATH  Google Scholar 

  • Schepsmeier U, Stöber J (2014) Derivatives and Fisher information of bivariate copulas. Stat Pap 55:525–542

    MathSciNet  MATH  Google Scholar 

  • Schepsmeier U, Stöber J, Brechmann et al (2015) R VineCopula: statistical inference of vine copulas, version 1.5, CRAN

  • Staplin ND, Kimber AC, Collett D, Roderick PJ (2015) Dependent censoring in piecewise exponential survival models. Stat Methods Med Res 24(3):325–341

    MathSciNet  Google Scholar 

  • Strzalkowska-Kominiak E, Stute W (2013) Empirical copulas for consecutive survival data. TEST 22:688–714

    MathSciNet  MATH  Google Scholar 

  • Tsai WY (1990) Testing the assumption of independence of truncation time and failure time. Biometrika 77:169–177

    MathSciNet  MATH  Google Scholar 

  • Wang MC (1991) Nonparametric estimation from cross-sectional survival data. J Am Stat Assoc 86(413):130–143

    MathSciNet  MATH  Google Scholar 

  • Woodroofe M (1985) Estimating a distribution function with truncated data. Ann Stat 13:163–177

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank two anonymous reviewers for their helpful comments that improved the paper. This work is supported by the Research Grants funded by the Government of Taiwan (MOST 103-2118-M-008-MY2; MOST 105-2118-M-008-003-MY2).

Author information

Authors and Affiliations

  1. Graduate Institute of Statistics, National Central University, No. 300, Zhongda Rd., Zhongli District, Taoyuan City, 32001, Taiwan

    Takeshi Emura & Chi-Hung Pan

Authors
  1. Takeshi Emura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chi-Hung Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takeshi Emura.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 1034 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Emura, T., Pan, CH. Parametric likelihood inference and goodness-of-fit for dependently left-truncated data, a copula-based approach. Stat Papers 61, 479–501 (2020). https://doi.org/10.1007/s00362-017-0947-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-017-0947-z

Keywords

Search

Navigation