Modified Weibull Distribution (MWD)

MWD Distribution

The modified Weibull distribution (MWD), introduced by Lai et al. (2003), which has been widely used in reliability and survival analysis. A random variable $X$ is said to follow a modified Weibull distribution if its cumulative distribution function $F(x)$ and probability density function $f(x)$ are given by $$F(x) = 1- \exp \big( -a x^b e^{\lambda x} \big),$$ and $$f(x) = a (b + \lambda x) x^{b-1} e^{\lambda x} \exp \big( -a x^b e^{\lambda x} \big),$$ where $x>0$, $a>0$ is the scale parameter, $b \ge 0$ is a shape parameter, and $\lambda \ge 0$ is an acceleration or flexibility parameter that controls how quickly the hazard grows over time. Then, the hazard function is $$h(x) = a (b + \lambda x) x^{b-1} e^{\lambda x}.$$ When $\lambda=0$, it reduces to the two-parameter Weibull distribution with $F(x) = 1- \exp(-a x^b)$. When $b=0$, it reduces to a type I extreme-value (or log-gamma) distribution with $F(x) = 1- \exp(-a e^{\lambda x} )$.

References

Lai, C. D., Xie, M., & Murthy, D. N. P. (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52(1), 33–37. doi.org/10.1109/TR.2002.805788