Flexible Weibull extension distribution
This distribution was proposed by Bebbington (2007). The probability density function \(f(x)\) and cumulative density function \(F(x)\) are given by:
\[f(x) = \left( \mu+ \frac{\sigma}{x^2} \right) e^{\mu x - \sigma / x} \exp \left( -e^{\mu x - \sigma / x} \right),\]
and
\[F(x) = 1 - \exp[-e^{\mu x - \sigma / x}], \quad x > 0.\]
respectively, where \(\mu > 0\), \(\sigma > 0\) and \(x > 0\).
Next figure shows possible shapes of the \(f(x)\) and \(F(x)\) for several values of the parameters.
Bebbington, M., C. D. Lai, and R. Zitikis. 2007. “A Flexible
Weibull Extension.” Reliability Engineering
& System Safety 92 (6): 719–26.