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The Kumaraswamy Weibull distribution with application to failure data. (English) Zbl 1202.62018
Summary: We introduce and study some mathematical properties of the P. Kumaraswamy [J. Hydrol. 46, 79–88 (1980)] Weibull distribution that is a quite flexible model in analyzing positive data. It contains as special sub-models the exponentiated Weibull, exponentiated Rayleigh, exponentiated exponential, Weibull and also the new Kumaraswamy exponential distribution. We provide explicit expressions for the moments and moment generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and Rényi entropy. The moments of the order statistics are calculated. We also discuss the estimation of the parameters by maximum likelihood. We obtain the expected information matrix. We provide applications involving two real data sets on failure times. Finally, some multivariate generalizations of the Kumaraswamy Weibull distribution are discussed.

MSC:
62E10 Characterization and structure theory of statistical distributions
62E20 Asymptotic distribution theory in statistics
62N05 Reliability and life testing
62E15 Exact distribution theory in statistics
62F10 Point estimation
62N02 Estimation in survival analysis and censored data
Software:
SPLIDA
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[1] Kumaraswamy, P., Generalized probability density-function for double-bounded random-processes, Journal of hydrology, 46, 79-88, (1980)
[2] Jones, M.C., Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages, Statistical methodology, 6, 70-91, (2009) · Zbl 1215.60010
[3] G.M. Cordeiro, M. de Castro, A new family of generalized distributions. Journal of Statistical Computation and Simulation (2010). DOI: 10.1080/0094965YY. · Zbl 1219.62022
[4] Muraleedharan, G.; Rao, A.D.; Kurup, P.G.; Unnikrishnan Nair, N.; Sinha, M., Modified Weibull distribution for maximum and significant wave height simulation and prediction, Coastal engineering, 54, 630-638, (2008)
[5] Eugene, N.; Lee, C.; Famoye, F., Beta-normal distribution and its applications, Communications in statistics—theory and methods, 31, 497-512, (2002) · Zbl 1009.62516
[6] Mudholkar, G.S.; Srivastava, D.K., Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE transactions on reliability, 42, 299-302, (1993) · Zbl 0800.62609
[7] Mudholkar, G.S.; Srivastava, D.K.; Freimer, M., The exponentiated Weibull family: a reanalysis of the bus-motor-failure data, Technometrics, 37, 436-445, (1995) · Zbl 0900.62531
[8] Cancho, V.; Bolfarine, H.; Achcar, J.A., A Bayesian analysis for the exponentiated-Weibull distribution, Journal applied statistical science, 8, 227-242, (1999) · Zbl 0924.62029
[9] Jiang, R.; Murthy, D.N.P., The exponentiated Weibull family: a graphical approach, IEEE transactions on reliability, 48, 68-72, (1999)
[10] Gupta, R.C.; Gupta, R.D.; Gupta, P.L., Modeling failure time data by lehman alternatives, Communications in statistics—theory and methods, 27, 887-904, (1998) · Zbl 0900.62534
[11] Gupta, R.D.; Kundu, D., Exponentiated exponential distribution: an alternative to gamma and Weibull distributions, Biometrical journal, 43, 117-130, (2001) · Zbl 0997.62076
[12] Nadarajah, S.; Gupta, A.K., A generalized gamma distribution with application to drought data, Mathematics and computers in simulation, 74, 1-7, (2007) · Zbl 1108.60011
[13] Ortega, E.M.M.; Bolfarine, H.; Paula, G.A., Influence diagnostics in generalized log-gamma regression models, Computational statistics and data analysis, 42, 165-186, (2003) · Zbl 1429.62336
[14] Wright, E.M., The asymptotic expansion of the generalized hypergeometric function, Journal of the London mathematical society, 10, 286-293, (1935) · Zbl 0013.02104
[15] Gradshteyn, I.S.; Ryzhik, I.M., Tableof integrals, series, and products, () · Zbl 0918.65002
[16] Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I., Integrals and series, vol. 1, (1986), Gordon and Breach Science Publishers Amsterdam · Zbl 0733.00004
[17] Leadbetter, M.R.; Lindgren, G.; Rootzén, H., Extremes and related properties of random sequences and process, (1987), Springer Verlag New York
[18] Barakat, H.M.; Abdelkader, Y.H., Computing the moments of order statistic from nonidentically random variables, Statistical methods and applications, 13, 15-26, (2004) · Zbl 1056.62012
[19] J.R.M. Hosking, The theory of probability weighted moments, Research Report RC12210, IBM Thomas J. Watson Research Center, New York, 1986.
[20] Meeker, W.Q.; Escobar, L.A., Statistical methods for reliability data, (1998), John Wiley New York · Zbl 0949.62086
[21] Murthy, D.N.P.; Xie, M.; Jiang, R., Weibull models, (2005), John Wiley New York
[22] Hougaard, P., A class of multivariate failure time distributions, Biometrika, 73, 671-678, (1986) · Zbl 0613.62121
[23] Lu, J.C.; Bhattacharyya, G.K., Some new constructions of bivariate Weibull models, Annals of the institute of statistical mathematics, 42, 543-559, (1990) · Zbl 0712.62094
[24] Mudholkar, G.S.; Hutson, A.D., The exponentiated Weibull family: some properties and a flood data application, Communication in statistics—theory and methods, 25, 3059-3083, (1996) · Zbl 0887.62019
[25] Nassar, M.M.; Eissa, F.H., On the exponentiated Weibull distribution, Communication in statistics—theory and methods, 32, 1317-1336, (2003) · Zbl 1140.62308
[26] Nadarajah, S.; Gupta, A.K., On the moments of the exponentiated Weibull distribution, Communication in statistics—theory and methods, 35, 253-256, (2005) · Zbl 1137.62308
[27] Choudhury, A., A simple derivation of moments of the exponentiated Weibull distribution, Metrika, 62, 17-22, (2005) · Zbl 1079.62023
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