---
title: "Modified Weibull Distribution (MWD)"
# author: "Fatih Kızılaslan"
# date: "`r format(Sys.time(), '%B %d, %Y')`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Modified Weibull Distribution (MWD)}
  %\VignetteEngine{knitr::rmarkdown}
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---

## MWD Distribution 

The modified Weibull distribution (MWD), introduced by [Lai et al. (2003)](https://doi.org/10.1109/TR.2002.805788), 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](https://doi.org/10.1109/TR.2002.805788)