The Quasi XGamma Poisson distribution

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

Sen, S., Korkmaz, M. Ç., & Yousof, H. M. (2018). The quasi XGamma-Poisson distribution: properties and application. Istatistik Journal of The Turkish Statistical Association, 11(3), 65-76.

Author

Simon Zapata

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