Skip to main content
SpringerLink
Log in

New robust tests for the parameters of the Weibull distribution for complete and censored data

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

Using the likelihood depth, new consistent and robust tests for the parameters of the Weibull distribution are developed. Uncensored as well as type-I right-censored data are considered. Tests are given for the shape parameter and also the scale parameter of the Weibull distribution, where in each case the situation that the other parameter is known as well the situation that both parameter are unknown is examined. In simulation studies the behavior in finite sample size and in contaminated data is analyzed and the new method is compared to existing ones. Here it is shown that the new tests based on likelihood depth give quite good results compared to standard methods and are robust against contamination. They are also robust in right-censored data in contrast to existing methods like the method of medians.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Balakrishnan N, Stehlík M (2008) Exact likelihood ratio test of the scale for censored Weibull sample. IFAS Research Paper Series 35

  • Boudt K, Caliskan D, Croux C (2011) Robust and explicit estimators for Weibull parameters. Metrika 73:187–209

    Article  MATH  MathSciNet  Google Scholar 

  • Chen Z (1997) Statistical inference about the shape parameter of the Weibull distribution. Stat Probab Lett 36:85–90

    Article  MATH  Google Scholar 

  • Coakley CW, Hettmansperger TP (1994) The maximum resistance of tests. Aust J Stat 36:225–233

    Article  MATH  MathSciNet  Google Scholar 

  • Denecke L (2010) Estimators and tests based on likelihood-depth with application to Weibull distribution, Gaussian and Gumbel copula. Ph. D. thesis, University of Kassel, Germany, urn:nbn:de:hebis:34–2010101234699

  • Denecke L, Müller ChH (2011) Robust estimators and tests for bivariate copulas based on likelihood depth. Comput Stat Data Anal 55:2724–2738

    Article  Google Scholar 

  • Denecke L, Müller ChH (2012) Consistency and robustness of tests and estimators based on depth. J Statist Plann Inference 142:2501–2517

    Article  MATH  MathSciNet  Google Scholar 

  • He X, Fung WK (1999) Method of medians for lifetime data with Weibull models. Stat Med 18:1993–2009

    Article  Google Scholar 

  • He X, Simpson DG, Portnoy SL (1990) Breakdown robustness of tests. J Am Stat Assoc 85:446–452

    Article  MATH  MathSciNet  Google Scholar 

  • Hoel DG (1972) A representation of mortality data by competing risks. Biometrics 28:475–488

    Article  Google Scholar 

  • Hu Y, Wang Y, Wu Y, Li Q, Hou C (2009) Generalized Mahalanobis depth in the reproducing kernel Hilbert space. Stat Pap 52:511–522

    Google Scholar 

  • Kahle W (1996) Estimation of the parameters of the Weibull distribution for censored samples. Metrika 44:27–40

    Article  MATH  MathSciNet  Google Scholar 

  • Kalbfleisch JD, Prentice RL (1980) The statistical analysis of failure time data. Wiley, New York

    MATH  Google Scholar 

  • Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley Series in Probability and Statistics, New York

    MATH  Google Scholar 

  • Lee AJ (1990) \(U\)-Statistics. Theory and practice. Marcel Dekker, New York

    MATH  Google Scholar 

  • Li J, Liu RY (2008) Multivariate spacings based on data depth: I. Construction of nonparametric multivariate tolerance regions. Ann Stat 36:1299–1323

    Google Scholar 

  • Lin L, Chen M (2006) Robust estimating equation based on statistical depth. Stat Pap 47:263–278

    Article  MATH  Google Scholar 

  • Liu RY (1988) On a notion of simplicial depth. Proc Natl Acad Sci USA 85:1732–1734

    Article  MATH  Google Scholar 

  • Liu RY (1990) On a notion of data depth based on random simplices. Ann Stat 18:405–414

    Article  MATH  Google Scholar 

  • López-Pintado S, Romo J (2009) On the concept of depth for functional data. J Am Stat Assoc 104:718–734

    Article  Google Scholar 

  • Mizera I, Müller ChH (2004) Location-scale depth. J Am Stat Assoc 99:949–989

    Article  MATH  Google Scholar 

  • Mizera I (2002) On depth and deep points: a calculus. Ann Stat 30:1681–1736

    Article  MATH  MathSciNet  Google Scholar 

  • Mosler K (2002) Multivariate dispersion, central regions and depth, the lift zonoid approach. Lecture Notes in Statistics 165, Springer, New York

  • Müller ChH (1997) Robust planning and analysis of experiments. Lecture Notes in Statistics 124, Springer, New York

  • Müller ChH (2005) Depth estimators and tests based on the likelihood principle with applications to regression. J Multivar Anal 95:153–181

    Article  MATH  Google Scholar 

  • R Core Team (2012) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna

    Google Scholar 

  • Rinne H (2009) The Weibull distribution—a handbook. CRC Press, London

    MATH  Google Scholar 

  • Romanazzi M (2009) Data depth, random simplices and multivariate dispersion. Stat Probab Lett 79:1473–1479

    Google Scholar 

  • Rousseeuw PJ, Hubert M (1999) Regression depth (with discussion). J Am Stat Assoc 94:388–433

    Article  MATH  MathSciNet  Google Scholar 

  • Tukey JW (1975) Mathematics and the picturing of data. In: Proceeding of the International Congress of Mathematicians 2. Vancouver, pp. 523–531 (1974)

  • Ylvisaker D (1977) Test resistance. J Am Stat Assoc 72:551–557

    Article  MathSciNet  Google Scholar 

  • Weibull W (1951) A statistical distribution function of wide applicability. ASME J Appl Mech 293–297

  • Wellmann R, Müller ChH (2010a) Tests for multiple regression based on simplicial depth. J Multivar Anal 101:824–838

    Article  MATH  Google Scholar 

  • Wellmann R, Müller ChH (2010b) Depth notions for orthogonal regression. J Multivar Anal 101:2358–2371

    Google Scholar 

  • Wellmann R, Harmand P, Müller, ChH (2009) Distribution-free tests for polynomial regression based on simplicial depth. J Multivar Anal 100:622–635

    Google Scholar 

  • Wong PG, Wong SP (1982) A curtailed test for the shape parameter of the Weibull distribution. Metrika 29:203–209

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang J (1996) The sample breakdown points of tests. J Stat Plan Inference 52:161–181

    Article  MATH  Google Scholar 

  • Zou Y, Serfling R (2000a) General notions of statistical depth functions. Ann Stat 28:461–482

    Article  Google Scholar 

  • Zou Y, Serfling R (2000b) Structural properties and convergence results for contours of sample statistical depth functions. Ann Stat 28:483–499

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referees for their helpful comments and valuable suggestions.

Author information

Authors and Affiliations

  1. Technische Universität Dortmund, Dortmund, Germany

    Liesa Denecke & Christine H. Müller

Authors
  1. Liesa Denecke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christine H. Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liesa Denecke.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Denecke, L., Müller, C.H. New robust tests for the parameters of the Weibull distribution for complete and censored data. Metrika 77, 585–607 (2014). https://doi.org/10.1007/s00184-013-0454-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-013-0454-8

Keywords

Search

Navigation