Density, distribution function,quantile function,
random generation and hazard function for the Quasi XGamma Poisson distribution
with parameters mu, sigma and nu.
Usage
dQXGP(x, mu, sigma, nu, log = FALSE)
pQXGP(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qQXGP(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rQXGP(n, mu, sigma, nu)
hQXGP(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.
Value
dQXGP gives the density, pQXGP gives the distribution
function, qQXGP gives the quantile function, rQXGP
generates random deviates and hQXGP gives the hazard function.
Details
The Quasi XGamma Poisson distribution with parameters mu,
sigma and nu has density given by:
\(f(x)= K(\mu, \sigma, \nu)(\frac {\sigma^{2} x^{2}}{2} + \mu) exp(\frac{\nu exp(-\sigma x)(1 + \mu + \sigma x + \frac {\sigma^{2}x^{2}}{2})}{1+\mu} - \sigma x),\)
for \(x > 0\), \(\mu> 0\), \(\sigma> 0\), \(\nu> 1\).
where
\(K(\mu, \sigma, \nu) = \frac{\nu \sigma}{(exp(\nu)-1)(1+\mu)}\)
References
Subhradev S, Mustafa C K, Haitham M Y (2018). “The Quasi XGamma-Poisson distribution: Properties and Application.” Istatistik: Journal of the Turkish Statistical Assocation, 11(3), 65--76. ISSN 1300-4077, https://dergipark.org.tr/en/pub/ijtsa/issue/42850/518206.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dQXGP(x, mu=0.5, sigma=1, nu=1), from=0.1, to=8,
ylim=c(0, 0.6), col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pQXGP(x, mu=0.5, sigma=1, nu=1),
from=0.1, to=8, col="red", las=1, ylab="F(x)")
curve(pQXGP(x, mu=0.5, sigma=1, nu=1, lower.tail=FALSE),
from=0.1, to=8, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qQXGP(p, mu=0.5, sigma=1, nu=1), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pQXGP(x, mu=0.5, sigma=1, nu=1),
from=0.1, add=TRUE, col="red")
## The random function
hist(rQXGP(n=1000, mu=0.5, sigma=1, nu=1), freq=FALSE,
xlab="x", ylim=c(0, 0.4), las=1, main="", xlim=c(0, 15))
curve(dQXGP(x, mu=0.5, sigma=1, nu=1),
from=0.001, to=500, add=TRUE, col="red")
## The Hazard function
curve(hQXGP(x, mu=0.5, sigma=1, nu=1), from=0.01, to=3,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
