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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 \(F(x) = 1- \exp(-a e^{\lambda x} )\).
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
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