The Sarhan and Zaindin's Modified Weibull distribution

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

Usage

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

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

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

rSZMW(n, mu, sigma, nu)

hSZMW(x, mu, sigma, nu)

Arguments

x, q: vector of quantiles.
mu: scale parameter.
sigma: shape parameter.
nu: shape 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

dSZMW gives the density, pSZMW gives the distribution function, qSZMW gives the quantile function, rSZMW generates random deviates and hSZMW gives the hazard function.

Details

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

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

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

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, Zaindin M (2009). “Modified Weibull distribution.” APPS. Applied Sciences, 11, 123–136.

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(dSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, to = 2, 
      ylim = c(0, 1.7), col = "red", las = 1, ylab = "f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, to = 2, ylim = c(0, 1),
      col = "red", las = 1, ylab = "F(x)")
curve(pSZMW(x, mu = 2, sigma = 1.5, nu = 0.2, lower.tail = FALSE), from = 0,
      to = 2, 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 = qSZMW(p = p, mu = 2, sigma = 1.5, nu = 0.2), y = p, xlab = "Quantile",
     las = 1, ylab = "Probability")
curve(pSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, add = TRUE, col = "red")

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


## The Hazard function
par(mfrow=c(1,1))
curve(hSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, to = 3, ylim = c(0, 8),
      col = "red", ylab = "Hazard function", las = 1)


par(old_par) # restore previous graphical parameters