dMOK: The Marshall-Olkin Kappa distribution

In RelDists: Estimation for some Reliability Distributions

The Marshall-Olkin Kappa distribution

Description

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

Usage

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.

Details

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

f(x)=\frac{τ\frac{μν}{σ}≤ft(\frac{x}{σ}\right)^{ν-1} ≤ft(μ+≤ft(\frac{x}{σ}\right)^{μν}\right)^{-\frac{μ+1}{μ}}}{≤ft[τ+(1-τ)≤ft(\frac{≤ft(\frac{x}{σ}\right)^{μν}}{μ+≤ft(\frac{x}{σ}\right)^{μν}}\right)^{\frac{1}{μ}}\right]^2}

for x > 0.

Value

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

Author(s)

Angel Muñoz,

References

\insertRef

javed2018marshallRelDists

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

RelDists documentation built on Dec. 23, 2022, 1:26 a.m.