Skip to main content
SpringerLink
Log in

The Kumaraswamy Gumbel distribution

  • Published:
Statistical Methods & Applications Aims and scope Submit manuscript

Abstract

The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. We propose a generalization—referred to as the Kumaraswamy Gumbel distribution—and provide a comprehensive treatment of its structural properties. We obtain the analytical shapes of the density and hazard rate functions. We calculate explicit expressions for the moments and generating function. The variation of the skewness and kurtosis measures is examined and the asymptotic distribution of the extreme values is investigated. Explicit expressions are also derived for the moments of order statistics. The methods of maximum likelihood and parametric bootstrap and a Bayesian procedure are proposed for estimating the model parameters. We obtain the expected information matrix. An application of the new model to a real dataset illustrates the potentiality of the proposed model. Two bivariate generalizations of the model are proposed.

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

Similar content being viewed by others

References

  • Andrade NLR, Moura RMP, Silveira A (2007) Determinação da Q7,10 para o Rio Cuiabá, Mato Grosso, Brasil e comparação com a vazão regularizada após a implantação do reservatório de aproveitamento múltiplo de manso. 24º Congresso Brasileiro de Engenharia Sanitária e Ambiental. Belo Horizonte, Minas Gerais Brasil.

  • Barakat HM, Abdelkader YH (2004) Computing the moments of order statistic from nonidentically random variables. Stat Methods Appl 13: 15–26

    Article  MathSciNet  MATH  Google Scholar 

  • Chen G, Balakrishnan N (1995) A general purpose approximate goodness-of-fit test. J Qual Technol 27: 154–161

    Google Scholar 

  • Cordeiro GM, de Castro M (2011) A new family of generalized distributions. J Stat Comput Simul 81: 883–898

    Article  MathSciNet  MATH  Google Scholar 

  • Cordeiro GM, Ortega EMM, Nadarajah S (2010) The Kumaraswamy Weibull distribution with application to failure data. J Frankl Inst 347: 1399–1429

    Article  MathSciNet  MATH  Google Scholar 

  • Cowles MK, Carlin BP (1996) Markov chain Monte Carlo convergence diagnostics: a comparative review. J Am Stat Assoc 91: 133–169

    MathSciNet  Google Scholar 

  • Davison AC, Hinkley DV (1997) Bootstrap methods and their application. Cambridge University Press, New York

    MATH  Google Scholar 

  • DiCiccio TJ, Efron B (1996) Bootstrap confidence intervals. Stat Sci 11: 189–228

    Article  MathSciNet  MATH  Google Scholar 

  • Doornik JA (2007) An object-oriented matrix language Ox 5. Timberlake Consultants Press, London

    Google Scholar 

  • Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7: 1–26

    Article  MathSciNet  MATH  Google Scholar 

  • Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman and Hall, New York

    MATH  Google Scholar 

  • Eugene N, Lee C, Famoye F (2002) Beta-normal distribution and its applications. Commun Stat Theory Methods 31: 497–512

    Article  MathSciNet  MATH  Google Scholar 

  • Fletcher SC, Ponnambalam K (1996) Estimation of reservoir yield and storage distribution using moments analysis. J Hydrol 182: 259–275

    Article  Google Scholar 

  • Ganji A, Ponnambalam K, Khalili D (2006) Grain yield reliability analysis with crop water demand uncertainty. Stoch Environ Res Risk Assess 20: 259–277

    Article  MathSciNet  Google Scholar 

  • Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences (with discussion). Stat Sci 7: 457–472

    Article  Google Scholar 

  • Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc B 52: 105–124

    MathSciNet  MATH  Google Scholar 

  • Jones MC (2004) Families of distributions arising from distributions of order statistics (with discussion). Test 13: 1–43

    Article  MathSciNet  MATH  Google Scholar 

  • Jones MC (2009) Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages. Stat Methodol 6: 70–91

    Article  MathSciNet  MATH  Google Scholar 

  • Kotz S, Nadarajah S (2000) Extreme value distributions: theory and applications. Imperial College Press, London

    Book  MATH  Google Scholar 

  • Koutsoyiannis D, Xanthopoulos T (1989) On the parametric approach to unit hydrograph identification. Water Resour Manag 3: 107–128

    Article  Google Scholar 

  • Kumaraswamy P (1980) Generalized probability density-function for double-bounded random-processes. J Hydrol 46: 79–88

    Article  Google Scholar 

  • Leadbetter MR, Lindgren G, Rootzén H (1987) Extremes and related properties of random sequences and process. Springer, New York

    Google Scholar 

  • Nadarajah S (2008) On the distribution of Kumaraswamy. J Hydrol 348: 568–569

    Article  Google Scholar 

  • Nadarajah S, Kotz S (2004) The beta Gumbel distribution. Math Probl Eng 4: 323–332

    Article  MathSciNet  Google Scholar 

  • Ponnambalam K, Seifi A, Vlach J (2001) Probabilistic design of systems with general distributions of parameters. Int J Circuit Theory Appl 29: 527–536

    Article  MATH  Google Scholar 

  • Prudnikov AP, Brychkov YA, Marichev OI (1986) Integrals and series, vols 1, 2 and 3. Gordon and Breach Science Publishers, Amsterdam

    Google Scholar 

  • Seifi A, Ponnambalam K, Vlach J (2000) Maximization of manufacturing yield of systems with arbitrary distributions of component values. Ann Oper Res 99: 373–383

    Article  MathSciNet  MATH  Google Scholar 

  • Sundar V, Subbiah K (1989) Application of double bounded probability density-function for analysis of ocean waves. Ocean Eng 16: 193–200

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Estatística, Universidade Federal de Pernambuco, Recife, PE, 50740-540, Brazil

    Gauss M. Cordeiro

  2. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Saralees Nadarajah

  3. Departamento de Ciências Exatas, Universidade de São Paulo, Piracicaba, SP, 13418-900, Brazil

    Edwin M. M. Ortega

Authors
  1. Gauss M. Cordeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saralees Nadarajah
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edwin M. M. Ortega
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saralees Nadarajah.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cordeiro, G.M., Nadarajah, S. & Ortega, E.M.M. The Kumaraswamy Gumbel distribution. Stat Methods Appl 21, 139–168 (2012). https://doi.org/10.1007/s10260-011-0183-y

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10260-011-0183-y

Keywords

Search

Navigation