A five-parameter distribution so-called the beta modified Weibull distribution is defined and studied. The new distribution contains, as special submodels, several important distributions discussed in the literature, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among others. The new distribution can be used effectively in the analysis of survival data since it accommodates monotone, unimodal and bathtub-shaped hazard functions. We derive the moments and examine the order statistics and their moments. We propose the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. A real data set is used to illustrate the importance and flexibility of the new distribution.
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Aarset MV (1987) How to identify bathtub hazard rate. IEEE Trans Reliab 36: 106–108
Barakat HM, Abdelkader YH (2004) Computing the moments of order statistics from nonidentical random variables. Stat Methods Appl 13: 15–26
Bebbington M, Lai CD, Zitikis R (2007) A flexible Weibull extension. Reliab Eng Syst Saf 92: 719–726
Brown BW, Floyd MS, Levy LB (2002) The log F: a distribution for all seasons. Comput Stat 17: 47–58
Carrasco JMF, Ortega EMM, Cordeiro GM (2008) A generalized modified Weibull distribution for lifetime modeling. Comput Stat Data Anal 53: 450–462
Cox C (2008) The generalized F distribution: an umbrella for parametric survival analysis. Stat Med 27: 4301–4312
Cox C, Chu H, Schneider MF, Mũoz A (2007) Tutorial in biostatistics: parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Stat Med 26: 4352–4374
Doornik J (2007) Ox 5: object-oriented matrix programming language, 5th ed. Timberlake Consultants, London
Eugene N, Lee C, Famoye F (2002) Beta-normal distribution and its applications. Commun Stat Theory Methods 31: 497–512
Gradshteyn IS, Ryzhik IM (2000) Table of integrals, series, and products. Academic Press, New York
Gupta RD, Kundu D (1999) Generalized exponential distributions. Aust NZ J Stat 41: 173–188
Gupta RD, Kundu D (2001) Exponentiated exponential distribution: an alternative to gamma and Weibull distributions. Biomet J 43: 117–130
Gupta AK, Nadarajah S (2004) Handbook of beta distribution and its applications. Marcel Dekker, New York
Haupt E, Schabe H (1992) A new model for a lifetime distribution with bathtub shaped failure rate. Microelectron Reliab 32: 633–639
Hosking JRM (1986) The theory of probability weighted moments. Research Report RC12210, IBM Thomas J. Watson Research Center, New York.
Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B 52: 105–124
Jones MC (2004) Family of distributions arising from distribution of order statistics. Test 13: 1–43
Kundu D, Rakab MZ (2005) Generalized Rayleigh distribution: different methods of estimation. Comput Stat Data Anal 49: 187–200
Lai CD, Xie M, Murthy DNP (2003) A modified Weibull distribution. Trans Reliab 52: 33–37
Lee C, Famoye F, Olumolade O (2007) Beta-Weibull distribution: some properties and applications to censored data. J Mod Appl Stat Methods 6: 173–186
Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure-real data. IEEE Trans Reliab 42: 299–302
Mudholkar GS, Srivastava DK, Friemer M (1995) The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37: 436–445
Mudholkar GS, Srivastava DK, Kollia GD (1996) A generalization of the Weibull distribution with application to the analysis of survival data. J Am Stat Assoc 91: 1575–1583
Nadarajah S, Gupta AK (2004) The beta Fréchet distribution. Far East J Theor Stat 14: 15–24
Nadarajah S, Kotz S (2004) The beta Gumbel distribution. Math Prob Eng 10: 323–332
Nadarajah S, Kotz S (2006) The beta exponential distribution. Reliab Eng Syst Saf 91: 689–697
Nelson W (1990) Accelerated life testing: statistical models, data analysis and test plans. Wiley, New York
Pham H, Lai CD (2007) On recent generalizations of the Weibull distribution. IEEE Trans Reliab 56: 454–458
Rajarshi S, Rajarshi MB (1988) Bathtub distributions: a review. Commun Stat Theory Methods 17: 2521–2597
Wang FK (2000) A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliab Eng Syst Saf 70: 305–312
Xie M, Lai CD (1995) Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab Eng Syst Saf 52: 87–93
Xie M, Tang Y, Goh TN (2002) A modified Weibull extension with bathtub failure rate function. Reliab Eng Syst Saf 76: 279–285
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Silva, G.O., Ortega, E.M.M. & Cordeiro, G.M. The beta modified Weibull distribution. Lifetime Data Anal 16, 409–430 (2010). https://doi.org/10.1007/s10985-010-9161-1