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The Exponentiated Modifien Weibull Extension distribution

Source: R/dEMWEx.R
dEMWEx.Rd

Density, distribution function, quantile function, random generation and hazard function for the Exponentiated Modifien Weibull Extension distribution with parameters mu, sigma, nu and tau.

Usage

dEMWEx(x, mu, sigma, nu, tau, log = FALSE)

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

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

rEMWEx(n, mu, sigma, nu, tau)

hEMWEx(x, mu, sigma, nu, tau)

Arguments

x, q

vector of quantiles.

mu

parameter.

sigma

parameter.

nu

parameter.

tau

parameter.

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.

Value

dEMWEx gives the density, pEMWEx gives the distribution function, qEMWEx gives the quantile function, rEMWEx generates random deviates and hEMWEx gives the hazard function.

Details

The Exponentiated Modifien Weibull Extension Distribution with parameters mu, sigma, nu and tau has density given by

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

for \(x > 0\), \(\nu> 0\), \(\mu > 0\), \(\sigma> 0\) and \(\tau > 0\).

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 .

Sarhan AM, Apaloo J (2013). “Exponentiated modified Weibull extension distribution.” Reliability Engineering & System Safety, 112, 137–144.

Author

Johan David Marin Benjumea, johand.marin@udea.edu.co

Examples

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

## The probability density function 
curve(dEMWEx(x, mu = 49.046, sigma =3.148, nu=0.00005, tau=0.1), from=0, to=100,
      col = "red", las = 1, ylab = "f(x)")


## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pEMWEx(x, mu = (1/4), sigma =1, nu=1, tau=2), from = 0, to = 1, 
      ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pEMWEx(x, mu = (1/4), sigma =1, nu=1, tau=2, lower.tail = FALSE), 
      from = 0, to = 1, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")


## The quantile function
p <- seq(from = 0, to = 0.99999, length.out = 100)
plot(x = qEMWEx(p = p, mu = 49.046, sigma =3.148, nu=0.00005, tau=0.1), y = p, 
     xlab = "Quantile", las = 1, ylab = "Probability")
curve(pEMWEx(x, mu = 49.046, sigma =3.148, nu=0.00005, tau=0.1), from = 0, add = TRUE, 
      col = "red")

## The random function
hist(rEMWEx(1000, mu = (1/4), sigma =1, nu=1, tau=2), freq = FALSE, xlab = "x", 
     las = 1, main = "")
curve(dEMWEx(x, mu = (1/4), sigma =1, nu=1, tau=2),  from = 0, add = TRUE, 
      col = "red", ylim = c(0, 0.5))


## The Hazard function(
par(mfrow=c(1,1))
curve(hEMWEx(x, mu = 49.046, sigma =3.148, nu=0.00005, tau=0.1), from = 0, to = 80, 
      col = "red", ylab = "Hazard function", las = 1)


par(old_par) # restore previous graphical parameters