dMW {RelDists}

The Modified Weibull distribution

Description

Density, distribution function, quantile function, random generation and hazard function for the modified weibull distribution with parameters mu, sigma and nu.

Usage

dMW(x, mu, sigma, nu, log = FALSE)

pMW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

qMW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

rMW(n, mu, sigma, nu)

hMW(x, mu, sigma, nu)

Arguments

x, q

vector of quantiles.

mu

shape parameter one.

sigma

parameter two.

nu

scale parameter three.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

p

vector of probabilities.

n

number of observations.

Details

The modified weibull distribution with parameters mu, sigma and nu has density given by

f(x) = \mu (\sigma + \nu x) x^{\sigma - 1} \exp(\nu x) \exp(-\mu x^{\sigma} \exp(\nu x))

for x > 0, \mu > 0, \sigma \geq 0 and \nu \geq 0.

Value

dMW gives the density, pMW gives the distribution function, qMW gives the quantile function, rMW generates random deviates and hMW gives the hazard function.

Author(s)

Johan David Marin Benjumea, [e-mail]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Lai CD, Xie M, Murthy DNP (2003). “A modified Weibull distribution.” IEEE Transactions on reliability, 52(1), 33–37.

Examples

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

## The probability density function 
curve(dMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=2,
  ylim=c(0, 1.5), col="red", las=1, ylab="f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=2,
 col = "red", las=1, ylab="F(x)")
curve(pMW(x, mu=2, sigma=1.5, nu=0.2, lower.tail = FALSE), 
from=0, to=2, col="red", las=1, ylab ="R(x)")

## The quantile function
p <- seq(from=0, to=0.9999, length.out=100)
plot(x=qMW(p, mu=2, sigma=1.5, nu=0.2), y=p, xlab="Quantile",
 las=1, ylab="Probability")
curve(pMW(x, mu=2, sigma=1.5, nu=0.2), from=0, add=TRUE, col="red")

## The random function
hist(rMW(n=1000, mu=2, sigma=1.5, nu=0.2), freq=FALSE,
 xlab="x", las=1, main="")
curve(dMW(x, mu=2, sigma=1.5, nu=0.2), from=0, add=TRUE, col="red")

## The Hazard function
par(mfrow=c(1,1))
curve(hMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=1.5, ylim=c(0, 5),
 col="red", las=1, ylab="H(x)", las=1)

par(old_par) # restore previous graphical parameters