dMOK function

The Marshall-Olkin Kappa distribution

Desnsity, distribution function, quantile function, random generation and hazard function for the Marshall-Olkin Kappa distribution with parameters mu, sigma, nu and tau.

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

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

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

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

hMOK(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.

Returns

dMOK gives the density, pMOK gives the distribution function, qMOK gives the quantile function, rMOK generates random deviates and hMOK gives the hazard function.

Details

The Marshall-Olkin Kappa distribution with parameters mu, sigma, nu and tau has density given by:

$f(x)=\frac{\tau\frac{\mu\nu}{\sigma}\left(\frac{x}{\sigma}\right)^{\nu-1} \left(\mu+\left(\frac{x}{\sigma}\right)^{\mu\nu}\right)^{-\frac{\mu+1}{\mu}}}{\left[\tau+(1-\tau)\left(\frac{\left(\frac{x}{\sigma}\right)^{\mu\nu}}{\mu+\left(\frac{x}{\sigma}\right)^{\mu\nu}}\right)^{\frac{1}{\mu}}\right]^2}$

for x > 0.

Examples

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

## The probability density function
par(mfrow = c(1,1))
curve(dMOK(x = x, mu = 1, sigma = 3.5, nu = 3, tau = 2), from = 0, to = 15,
      ylab = 'f(x)', col = 2, las = 1)

## The cumulative distribution and the Reliability function

par(mfrow = c(1,2))
curve(pMOK(q = x, mu = 1, sigma = 2.5, nu = 3, tau = 2), from = 0, to = 10,
     col = 2, lwd = 2, las = 1, ylab = 'F(x)')
curve(pMOK(q = x, mu = 1, sigma = 2.5, nu = 3, tau = 2, lower.tail = FALSE), from = 0, to = 10,
      col = 2, lwd = 2, las = 1, ylab = 'R(x)')

## The quantile function
p <- seq(from = 0.00001, to = 0.99999, length.out = 100)
plot(x = qMOK(p = p, mu = 4, sigma = 2.5, nu = 3, tau = 2), y = p, xlab = 'Quantile',
     las = 1, ylab = 'Probability')
curve(pMOK(q = x, mu = 4, sigma = 2.5, nu = 3, tau = 2), from = 0, to = 15,
      add = TRUE, col = 2)

## The random function

hist(rMOK(n = 10000, mu = 1, sigma = 2.5, nu = 3, tau = 2), freq = FALSE,
     xlab = "x", las = 1, main = '', ylim = c(0,.3), xlim = c(0,20), breaks = 50)
curve(dMOK(x, mu = 1, sigma = 2.5, nu = 3, tau = 2), from = 0, to = 15, add = TRUE, col = 2)

## The Hazard function

par(mfrow = c(1,1))
curve(hMOK(x = x, mu = 1, sigma = 2.5, nu = 3, tau = 2), from = 0, to = 20,
      col = 2, ylab = 'Hazard function', las = 1)

par(old_par) # restore previous graphical parameters

References

Rdpack::insert_ref(key="javed2018marshall",package="RelDists")

Author(s)

Angel Muñoz,