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

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

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

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} \,

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*^[1] 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

↑ "System evolution and reliability of systems". Sysev (Belgium). 2010-01-01.

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". Commun Stat Theory Meth. 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. JSTOR 1269735.
Nassar, M.M.; Eissa, F.H. (2003). "On the exponentiated Weibull distribution". Commun Stat Theory Meth. 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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Exponentiated Weibull distribution

Background

References

Further reading