dKumIW function

The Kumaraswamy Inverse Weibull distribution

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

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

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

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

rKumIW(n, mu, sigma, nu)

hKumIW(x, mu, sigma, nu)

Arguments

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

Returns

dKumIW gives the density, pKumIW gives the distribution function, qKumIW gives the quantile function, rKumIW

generates random deviates and hKumIW gives the hazard function.

Details

The Kumaraswamy Inverse Weibull Distribution with parameters mu, sigma and nu has density given by

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

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

Examples

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

## The probability density function 
par(mfrow = c(1, 1))
curve(dKumIW(x, mu = 1.5, sigma=  1.5, nu = 1), from = 0, to = 8.5, 
      col = "red", las = 1, ylab = "f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pKumIW(x, mu = 1.5, sigma=  1.5, nu = 1), from = 0, to = 8.5, 
      ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pKumIW(x, mu = 1.5, sigma=  1.5, nu = 1, lower.tail = FALSE), 
      from = 0, to = 6, 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 = qKumIW(p=p, mu = 1.5, sigma=  1.5, nu = 10), y = p, 
     xlab = "Quantile", las = 1, ylab = "Probability")
curve(pKumIW(x, mu = 1.5, sigma=  1.5, nu = 10), from = 0, add = TRUE, 
      col = "red")

## The random function
hist(rKumIW(1000, mu = 1.5, sigma=  1.5, nu = 5), freq = FALSE, xlab = "x", 
     las = 1, ylim = c(0, 1.5), main = "")
curve(dKumIW(x, mu = 1.5, sigma=  1.5, nu = 5), from = 0, to =8, add = TRUE, 
      col = "red")

## The Hazard function
par(mfrow=c(1,1))
curve(hKumIW(x, mu = 1.5, sigma=  1.5, nu = 1), from = 0, to = 3, 
      ylim = c(0, 0.7), col = "red", ylab = "Hazard function", las = 1)

par(old_par) # restore previous graphical parameters

References

Rdpack::insert_ref(key="almalki2014modifications",package="RelDists")

Rdpack::insert_ref(key="shahbaz2012kumaraswamy",package="RelDists")

Author(s)

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