# Exponentiated Weibull distribution

The exponentiated Weibull distribution is generalization of the Weibull distribution. It was first introduced by Mudholkar and Srivastava in 1993 [1] as a flexible alternative to the Weibull distribution, especially for modeling lifetime data with unimodal hazard rates. The distribution is formed by exponentiating the Weibull cumulative distribution function (CDF), adding a second shape parameter for more flexibility. It has been used in various fields, including engineering, medicine, and social sciences, to model lifetime data.

Compared to the Weibull distribution, the EW distribution is more flexible in terms of the shapes it can accommodate for the hazard rate function., such as increasing, decreasing, bathtub-shaped, and unimodal hazard rates. This makes it a useful tool for modeling complex data sets, allowing for more accurate modeling of complex lifetime data. However, the additional flexibility can also make it more challenging to estimate the model parameters. Additionally, the EW distribution may require larger sample sizes than the Weibull distribution to estimate parameters accurately.

The negative exponential (NE) distribution is another choice for modeling lifetime data. In Mudholkar and Srivastava’s paper, they discuss how the EW distribution can reduce to the NE distribution under certain special cases. However, the EW distribution can accommodate non-monotonic hazard rates, which the NE distribution cannot.

## Exponentiated Weibull distribution properties

In the exponentiated Weibull (EW) distribution, the hazard rate function can take on a variety of shapes, including increasing, unimodal, and decreasing curves.

for x > 0, and F(xk; λ; α) = 0 for x < 0 and

•  k > 0 = first shape parameter.
• α > 0 = second shape parameter
• λ > 0 = scale parameter.

The probability density function (PDF) of the EW distribution is given by:

The Weibull distribution is a special case of the EW distribution. When α = 1, the EW distribution reduces to the Weibull distribution. When k = 1, the PDF gives the exponentiated exponential distribution.

## History and use of the EW distribution

The exponentiated Weibull (EW) distribution was primarily introduced as a flexible alternative to the Weibull distribution. Before the introduction of the EW distribution, many distributions could not accommodate non-monotonic hazard rates, and the existing distributions that could accommodate these hazard rates were not very flexible.

In survival analysis, the hazard rate is the probability of an event occurring in a small interval of time, given that the event hasn’t happened yet. For example, when studying the failure times of a machine, the hazard rate would be the instantaneous rate at which the machine is likely to fail at a given time, given that it has not already failed. A non-monotonic hazard rate is a hazard rate that does not exhibit a consistent trend of increasing or decreasing over time. It can be characterized by having a shape that is not monotonic (i.e., always increasing or always decreasing) but may instead vary over time in a non-uniform manner. For example, a bathtub-shaped hazard rate exhibits a higher hazard rate during its initial phase, followed by a relatively low hazard rate and then another high hazard rate toward the end of the study period. Such hazard rates can be challenging to model with standard distributions like the Weibull or exponential distributions, which assume that the hazard rate is either always increasing or always decreasing. That’s where distributions like the exponentiated Weibull come in, which can more flexibly accommodate non-monotonic hazard rates.

The EW distribution was first introduced by Mudholkar and Srivastava in 1993, and its flexibility compared to the Weibull distribution was noted. The family of distributions was later extended by different researchers, including Kus et al. [3].

The information about the different hazard rate functions accommodated by the EW distribution is essential as it highlights its flexibility and ability to model various data sets with non-monotonic hazard rates. This flexibility makes it a valuable distribution to use in many situations where other distributions may not provide adequate fitting. Additionally, the ability to account for different hazard rate functions can help in identifying and mitigating potential failure modes in systems, improving reliability, and reducing failures.

## References

[1] Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42(2), 299-302. https://doi.org/10.1109/24.229504

[2] Research on Laser Atmospheric Transmission Performance Based on Exponentiated Weibull Distribution Channel Model – Scientific Figure on ResearchGate. Available from: https://www.researchgate.net/figure/Comparison-of-exponentiated-Weibull-distribution-and-negative-exponential-distribution_fig1_289049297 [accessed 9 May, 2023]  Creative Commons Attribution 4.0 International

[3] Kus, C., Unal, G., & Cevik, A. S. (2008). The Exponentiated Weibull Distribution: A Survey. Hacettepe Journal of Mathematics and Statistics, 37(1), 79-91. https://dergipark.org.tr/tr/download/article-file/191632

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