Skip to main content
Springer Nature Link
Log in

New classes of tests for the Weibull distribution using Stein’s method in the presence of random right censoring

  • Original paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

We develop two new classes of tests for the Weibull distribution based on Stein’s method. The proposed tests are applied in the full sample case as well as in the presence of random right censoring. We investigate the finite sample performance of the new tests using a comprehensive Monte Carlo study. In both the absence and presence of censoring, it is found that the newly proposed classes of tests outperform competing tests against the majority of the distributions considered. In the cases where censoring is present we consider various censoring distributions. Some remarks on the asymptotic properties of the proposed tests are included. We present another result of independent interest; a test initially proposed for use with full samples is amended to allow for testing for the Weibull distribution in the presence of censoring. The techniques developed in the paper are illustrated using two practical examples.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

Availability of data and material

All data used are in the public domain.

References

  • Allison J, Milošević B, Obradović M, Smuts M (2021) Distribution-free goodness-of-fit tests for the pareto distribution based on a characterization. Comput Stat pp 1–16

  • Allison J, Santana L (2015) On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests. J Stat Comput Simul 85(16):3276–3288

    Article  MathSciNet  Google Scholar 

  • Allison JS, Betsch S, Ebner B, Visagie IJH (2019) New weighted L\(^2\)-type tests for the inverse Gaussian distribution. arXiv preprint arXiv:1910.14119

  • Allison JS, Huskova M, Meintanis SG (2017) Testing the adequacy of semiparametric transformation models. Test 27:1–25

    MathSciNet  MATH  Google Scholar 

  • Allison JS, Santana L, Smit N, Visagie IJH (2017) An “apples-to-apples” comparison of various tests for exponentiality. Comput Stat 32(4):1241–1283

  • Balakrishnan N, Chimitova E, Vedernikova M (2015) An empirical analysis of some nonparametric goodness-of-fit tests for censored data. Commun Stat Simul Comput 44(4):1101–1115

    Article  MathSciNet  Google Scholar 

  • Baringhaus L, Ebner B, Henze N (2017) The limit distribution of weighted L2-goodness-of-fit statistics under fixed alternatives, with applications. Ann Inst Stat Math 69(5):969–995

    Article  Google Scholar 

  • Baringhaus L, Henze N (1988) A consistent test for multivariate normality based on the empirical characteristic function. Metrika 35(1):339–348

    Article  MathSciNet  Google Scholar 

  • Baringhaus L, Henze N (1991) A class of consistent tests for exponentiality based on the empirical laplace transform. Ann Inst Stat Math 43(3):551–564

    Article  MathSciNet  Google Scholar 

  • Betsch S, Ebner B (2018) Testing normality via a distributional fixed point property in the Stein characterization. TEST pp. 1–34. https://doi.org/10.1007/s11749-019-00630-0

  • Betsch S, Ebner B (2019) A new characterization of the gamma distribution and associated goodness-of-fit tests. Metrika 82(7):779–806

    Article  MathSciNet  Google Scholar 

  • Bothma E, Allison JS, Cockeran M, Visagie IJH (2020) Kaplan-Meier based tests for exponentiality in the presence of censoring. arXiv preprint arXiv:2011.04519

  • Breslow N, Crowley J (1974) A large sample study of the life table and product limit estimates under random censorship. Ann Stat pp. 437–453

  • Cabaña A, Quiroz AJ (2005) Using the empirical moment generating function in testing for the Weibull and the type I extreme value distributions. Test 14(2):417–431

    Article  MathSciNet  Google Scholar 

  • Cox DR, Oakes D (1984) Analysis of survival data, vol 21. CRC Press, Boca Raton

    Google Scholar 

  • Cuparić M, Milošević B (2021) New characterization-based exponentiality tests for randomly censored data. TEST. https://doi.org/10.1007/s11749-021-00787-7

    Article  Google Scholar 

  • D’Agostino RB, Stephens MA (1986) Goodness-of-fit techniques, vol 68. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Döring M, Cramer E (2019) On the power of goodness-of-fit tests for the exponential distribution under progressive Type-II censoring. J Stat Comput Simul 89:2997–3034

    Article  MathSciNet  Google Scholar 

  • Efron B (1967) The two sample problem with censored data. In: Proceedings of the fifth berkeley symposium on mathematical statistics and probability, vol 4, pp 831–853

  • Fernández T, Rivera N (2020) Kaplan-Meier V-and U-statistics. Electron J Stat 14(1):1872–1916

    Article  MathSciNet  Google Scholar 

  • Fernández VA, Jiménez Gamero MD, García M (2008) A test for the two-sample problem based on empirical characteristic functions. Comput Stat Data Anal 52:3730–3748

    Article  MathSciNet  Google Scholar 

  • Feuerverger A, Mureika RA (1977) The empirical characteristic function and its applications. Ann Stat 5(1):88–97

    Article  MathSciNet  Google Scholar 

  • Giacomini R, Politis DN, White H (2013) A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econom Theory 29(3):567–589

    Article  MathSciNet  Google Scholar 

  • Gupta RD, Richards DSP (1997) Invariance properties of some classical tests for exponentiality. J Stat Plann Inference 63(2):203–213

    Article  MathSciNet  Google Scholar 

  • Henze N, Visagie IJH (2020) Testing for normality in any dimension based on a partial differential equation involving the moment generating function. Ann Inst Stat Math 72:1109–1136

    Article  MathSciNet  Google Scholar 

  • Hlavac M (2018) stargazer: well-formatted regression and summary statistics tables. https://CRAN.R-project.org/package=stargazer

  • Jacobson A, Wilson V, Pileggi S (2018) parmsurvfit: Parametric Models for Survival Data. https://CRAN.R-project.org/package=parmsurvfit. R package version 0.1.0

  • Jiang R, Murthy D (2011) A study of Weibull shape parameter: properties and significance. Reliab Eng Syst Saf 96(12):1619–1626

    Article  Google Scholar 

  • Kalbfleisch JD, Prentice RL (2011) The statistical analysis of failure time data, vol 360. Wiley, New York

    MATH  Google Scholar 

  • Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53(282):457–481

    Article  MathSciNet  Google Scholar 

  • Kim N (2017) Goodness-of-fit tests for randomly censored Weibull distributions with estimated parameters. Commun Stat Appl Methods 24(5):519–531

    Google Scholar 

  • Klar B, Meintanis SG (2005) Tests for normal mixtures based on the empirical characteristic function. Comput Stat Data Anal 49(1):227–242

    Article  MathSciNet  Google Scholar 

  • Kotz S, Nadarajah S (2000) Extreme value distributions: theory and applications. World Scientific, Singapore

    Book  Google Scholar 

  • Kotze S, Johnson N (1983) Encyclopedia of statistical sciences, vol 3. Wiley, New York

    Google Scholar 

  • Koziol JA, Green SB (1976) A Cramér-von Mises statistic for randomly censored data. Biometrika 63(3):465–474

    MathSciNet  MATH  Google Scholar 

  • Krit M (2014) Goodness-of-fit tests for the Weibull distribution based on the Laplace transform. Journal de la Société Française de Statistique 155(3):135–151

    MathSciNet  MATH  Google Scholar 

  • Lee ET, Wang J (2003) Statistical methods for survival data analysis, vol 476. Wiley, New York

    Book  Google Scholar 

  • Liao M, Shimokawa T (1999) A new goodness-of-fit test for type-i extreme-value and 2-parameter Weibull distributions with estimated parameters. Optimization 64(1):23–48

    MathSciNet  MATH  Google Scholar 

  • Mann NR, Scneuer EM, Fertig KW (1973) A new goodness-of-fit test for the two-parameter Weibull or extreme-value distribution with unknown parameters. Commun Stat Theory Methods 2(5):383–400

    MathSciNet  MATH  Google Scholar 

  • Mazucheli J, Fernandes LB, de Oliveira RP (2016) LindleyR: The Lindley Distribution and Its Modifications. https://CRAN.R-project.org/package=LindleyR. R package version 1.1.0

  • Meintanis SG, Iliopoulos G (2003) Tests of fit for the Rayleigh distribution based on the empirical Laplace transform. Ann Inst Stat Math 55(1):137–151

    MathSciNet  MATH  Google Scholar 

  • Meintanis SG, Ngatchou-Wandji J, Allison JS (2018) Testing for serial independence in vector autoregressive models. Stat Papers 59(4):1379–1410

    Article  MathSciNet  Google Scholar 

  • Mijburgh PA, Visagie IJH (2020) An overview of goodness-of-fit tests for the Poisson distribution. South African Stat J 54(2):207–230

    Article  Google Scholar 

  • R Core Team (2019) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

  • Tiku ML, Singh M (1981) Testing the two parameter Weibull distribution. Commun Stat Theory Methods 10(9):907–918

    Article  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Subject Group Statistics, North-West University, Potchefstroom, South Africa

    E. Bothma, J. S. Allison & I. J. H. Visagie

Authors
  1. E. Bothma
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. J. S. Allison
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. I. J. H. Visagie
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to I. J. H. Visagie.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Code availability

Custom code was written and are available upon request.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bothma, E., Allison, J.S. & Visagie, I.J.H. New classes of tests for the Weibull distribution using Stein’s method in the presence of random right censoring. Comput Stat 37, 1751–1770 (2022). https://doi.org/10.1007/s00180-021-01178-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-021-01178-0

Keywords