Exponentiated Weibull distribution

In statistics, the exponentiated Weibull family of probability distributions was introduced by Mudholkar and Srivastava (1993) as an extension of the Weibull family obtained by adding a second shape parameter. The cumulative distribution function for the exponentiated Weibull distribution is

[math]\displaystyle{ F(x;k,\lambda; \alpha) = \left[ 1- e^{-(x/\lambda)^k} \right]^\alpha \, }[/math]

for x > 0, and F(x; k; λ; α) = 0 for x < 0. Here k > 0 is the first shape parameter, α > 0 is the second shape parameter and λ > 0 is the scale parameter of the distribution.

The density is

[math]\displaystyle{ f(x;k,\lambda; \alpha) = \alpha \frac{k}{\lambda} \left[\frac{x}{\lambda}\right]^{k-1} \left[1- e^{-(x/\lambda)^k} \right]^{\alpha-1} e^{-(x/\lambda)^k} \, }[/math]

There are two important special cases:

• α = 1 gives the Weibull distribution;
• k = 1 gives the exponentiated exponential distribution.

Background

The family of distributions accommodates unimodal, bathtub shaped* and monotone failure rates. A similar distribution was introduced in 1984 by Zacks, called a Weibull-exponential distribution (Zacks 1984). Crevecoeur introduced it in assessing the reliability of ageing mechanical devices and showed that it accommodates bathtub shaped failure rates (1993, 1994). Mudholkar, Srivastava, and Kollia (1996) applied the generalized Weibull distribution to model survival data. They showed that the distribution has increasing, decreasing, bathtub, and unimodal hazard functions. Mudholkar, Srivastava, and Freimer (1995), Mudholkar and Hutson (1996) and Nassar and Eissa (2003) studied various properties of the exponentiated Weibull distribution. Mudholkar et al. (1995) applied the exponentiated Weibull distribution to model failure data. Mudholkar and Hutson (1996) applied the exponentiated Weibull distribution to extreme value data. They showed that the exponentiated Weibull distribution has increasing, decreasing, bathtub, and unimodal hazard rates. The exponentiated exponential distribution proposed by Gupta and Kundu (1999, 2001) is a special case of the exponentiated Weibull family. Later, the moments of the EW distribution were derived by Choudhury (2005). Also, M. Pal, M.M. Ali, J. Woo (2006) studied the EW distribution and compared it with the two-parameter Weibull and gamma distributions with respect to failure rate.

References

• Choudhury, A. (2005). "A Simple Derivation of Moments of the Exponentiated Weibull Distribution". Metrika 62 (1): 17–22. doi:10.1007/s001840400351. 
• Crevecoeur, G.U. (1993). "A model for the Integrity Assessment of Ageing Repairable Systems". IEEE Transactions on Reliability 42 (1): 148–155. doi:10.1109/24.210287. 
• Crevecoeur, G.U. (1994). "Reliability assessment of ageing operating systems". European Journal of Mechanical Engineering 39 (4): 219–228. 
• Liu, J.; Wang, Y. (2013). "On Crevecoeur's bathtub-shaped failure rate model". Computational Statistics & Data Analysis 57 (1): 645–660. doi:10.1016/j.csda.2012.08.002. 
• Mudholkar, G.S.; Hutson, A.D. (1996). "The exponentiated Weibull family: some properties and a flood data application". Communications in Statistics - Theory and Methods 25: 3059–3083. doi:10.1080/03610929608831886. 
• Mudholkar, G.S.; Srivastava, D.K. (1993). "Exponentiated Weibull family for analyzing bathtub failure-ratedata". IEEE Transactions on Reliability 42 (2): 299–302. doi:10.1109/24.229504. 
• Mudholkar, G.S.; Srivastava, D.K.; Freimer, M. (1995). "The exponentiated Weibull family; a reanalysis of the bus motor failure data". Technometrics 37 (4): 436–445. doi:10.2307/1269735. 
• Nassar, M.M.; Eissa, F.H. (2003). "On the exponentiated Weibull distribution". Communications in Statistics - Theory and Methods 32: 1317–1336. doi:10.1081/STA-120021561. 
• Pal, M.; Ali, M.M.; Woo, J. (2006). "Exponentiated Weibull distribution". Statistica 66 (2): 139–147. 
• Zacks, S. (1984). "Estimating the Shift to Wear-Out of Systems Having Exponential-Weibull Life Distributions". Operations Research 32 (3): 741–749. doi:10.1287/opre.32.3.741. 

Further reading

• Nadarajah, S.; Gupta, A.K. (2005). "On the Moments of the Exponentiated Weibull Distribution". Communications in Statistics - Theory and Methods 34 (2): 253–256. doi:10.1081/STA-200047460.