Generalized Weibull distribution

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

Usage

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

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

qGWF(p, mu, sigma, nu)

rGWF(n, mu, sigma, nu)

hGWF(x, mu, sigma, nu)

Arguments

x, q: vector of quantiles.
mu: parameter one.
sigma: parameter two.
nu: 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.

Value

dGWF gives the density, pGWF gives the distribution function, qGWF gives the quantile function, rGWF generates random deviates and hGWF gives the hazard function.

Details

The generalized Weibull with parameters mu, sigma and nu has density given by

$$f(x) = \mu \sigma x^{\sigma - 1} \left( 1 - \mu \nu x^\sigma \right)^{\frac{1}{\nu} - 1}$$

for $x > 0$, $\mu>0$, $\sigma>0$ and $-\infty<\nu<\infty$.

References

Mudholkar, G. S., & Kollia, G. D. (1994). Generalized Weibull family: a structural analysis. Communications in statistics-theory and methods, 23(4), 1149-1171.

Author

Jaime Mosquera, jmosquerag@unal.edu.co

Examples

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

## The probability density function
curve(
    dGWF(x, mu = 5, sigma = 2, nu = -0.2),
    from = 0, to = 5, col = "red", las = 1, ylab = "f(x)"
)


## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(
    pGWF(x, mu = 5, sigma = 2, nu = -0.2),
    from = 0, to = 5, ylim = c(0, 1),
    col = "red", las = 1, ylab = "F(x)"
)
curve(
    pGWF(
        x, mu = 5, sigma = 2, nu = -0.2, 
        lower.tail = FALSE
    ),
    from = 0, to = 5, ylim = c(0, 1),
    col = "red", las = 1, ylab = "R(x)"
)


## The quantile function
p <- seq(from = 0, to = 0.999, length.out = 100)
plot(
    x = qGWF(p, mu = 5, sigma = 2, nu = -0.2),
    y = p, xlab = "Quantile", las = 1,
    ylab = "Probability"
)
curve(
    pGWF(x, mu = 5, sigma = 2, nu = -0.2),
    from = 0, add = TRUE, col = "red"
)

## The random function
hist(
    rGWF(n = 10000, mu = 5, sigma = 2, nu = -0.2),
    freq = FALSE, xlab = "x", las = 1, main = "", ylim = c(0, 2.0)
)
curve(dGWF(x, mu = 5, sigma = 2, nu = -0.2),
    from = 0, add = TRUE, col = "red"
)


## The Hazard function
par(mfrow = c(1, 1))
curve(
    hGWF(x, mu = 0.003, sigma = 5e-6, nu = 0.025),
    from = 0, to = 250, col = "red", 
    ylab = "Hazard function", las = 1
)


par(old_par) # restore previous graphical parameters