FWE distribution"
In RelDists: Estimation for some Reliability Distributions

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(RelDists)

Flexible Weibull extension distribution

This distribution was proposed by Bebbington [-@Bebbington2007]. 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.

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

paleta <- c("#00004A", "#00A4FF", "#F6F906", "#FF3300")
par(mfrow=c(1, 2))

# bathtub-like shape
m <- 1/8
s1 <- 3/8
s2 <- 9/8
s3 <- 27/8
s4 <- 81/8

curve(dFWE(x, mu=m, sigma=s1), from=0, to=15, ylim=c(0, 1), 
      col=paleta[1], ylab="f(x)", las=1, lwd=2)
curve(dFWE(x, mu=m, sigma=s2), col=paleta[2], add=T, lwd=2)
curve(dFWE(x, mu=m, sigma=s3), col=paleta[3], add=T, lwd=2)
curve(dFWE(x, mu=m, sigma=s4), col=paleta[4], add=T, lwd=2)

cap1 <- as.expression(bquote(mu * ' = ' * .(m) * ', ' * sigma * ' = ' * .(s1)))
cap2 <- as.expression(bquote(mu * ' = ' * .(m) * ', ' * sigma * ' = ' * .(s2)))
cap3 <- as.expression(bquote(mu * ' = ' * .(m) * ', ' * sigma * ' = ' * .(s3)))
cap4 <- as.expression(bquote(mu * ' = ' * .(m) * ', ' * sigma * ' = ' * .(s4)))

legend('topright', legend=c(cap1, cap2, cap3, cap4), 
       col=paleta, lty=c(1, 1, 1), bty="n", lwd=2)

# shallower bathtup-like shape
s <- 9/8
m1 <- 1/8
m2 <- 2/8
m3 <- 3/8
m4 <- 4/8

curve(pFWE(x, mu=m1, sigma=s), from=0, to=10, ylim=c(0, 1), 
      col=paleta[1], ylab="F(x)", las=1, lwd=2)
curve(pFWE(x, mu=m2, sigma=s), col=paleta[2], add=T, lwd=2)
curve(pFWE(x, mu=m3, sigma=s), col=paleta[3], add=T, lwd=2)
curve(pFWE(x, mu=m4, sigma=s), col=paleta[4], add=T, lwd=2)

cap1 <- as.expression(bquote(mu * ' = ' * .(m1) * ', ' * sigma * ' = ' * .(s)))
cap2 <- as.expression(bquote(mu * ' = ' * .(m2) * ', ' * sigma * ' = ' * .(s)))
cap3 <- as.expression(bquote(mu * ' = ' * .(m3) * ', ' * sigma * ' = ' * .(s)))
cap4 <- as.expression(bquote(mu * ' = ' * .(m4) * ', ' * sigma * ' = ' * .(s)))

legend('bottomright', legend=c(cap1, cap2, cap3, cap4), 
       col=paleta, lty=c(1, 1, 1), bty="n", lwd=2)

par(old_par) # restore previous graphical parameters