Use of Lambert W function in the simulation of Weibull and non-Weibull distributions

TL;DR

The paper tackles the problem of simulating distributions whose inverse CDFs lack closed forms by employing the Lambert $W$ function to transform quantile equations into solvable forms. It demonstrates closed-form or highly accurate Lambert $W$–based quantile expressions for a range of Weibull models (including two-, three-, four-, and five-parameter variants) and provides approximations for several intractable cases. The work also extends the Lambert $W$ framework to non-Weibull lifetime distributions, offering practical quantile-based random variate generation methods. Overall, the approach broadens the set of distributions amenable to Monte Carlo simulation by enabling direct quantile-based sampling where traditional inversion fails, with clear formulas and open problems highlighted for further research.

Abstract

In this work, we have taken up some distributions, mostly Weibull family, whose quantile functions could not be obtained using the traditional inversion method. We have solved the same quantile functions by using the inversion method only, with the additional help of transcendental functions like the Lambert W function. The usage of the Lambert W function has helped derive closed-form solutions for mathematical models in many fields, for which explicit or exact solutions were not available and approximation was the only known way of approaching it.