The Quasi XGamma Poisson family

Usage

QXGP(mu.link = "log", sigma.link = "log", nu.link = "log")

Arguments

mu.link: defines the mu.link, with "log" link as the default for the mu parameter.
sigma.link: defines the sigma.link, with "log" link as the default for the sigma.
nu.link: defines the nu.link, with "log" link as the default for the nu parameter.

Value

Returns a gamlss.family object which can be used to fit a QXGP distribution in the gamlss() 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

Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rQXGP(n=200, mu=4, sigma=2, nu=3)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='QXGP',
              control=gamlss.control(n.cyc=5000, trace=FALSE))

# Extracting the fitted values for mu, sigma and nu
# using the inverse link function
exp(coef(mod, what='mu'))
#> (Intercept) 
#>      269437 
exp(coef(mod, what='sigma'))
#> (Intercept) 
#>    1.490796 
exp(coef(mod, what='nu'))
#> (Intercept) 
#>    2.939705 

# Example 2
# Generating random values under some model
n <- 2000
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-2.19 + 3 * x1)
sigma <- exp(1 - 2 * x2)
nu <- 1
x <- rQXGP(n=n, mu, sigma, nu)

mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=QXGP,
              control=gamlss.control(n.cyc=5000, trace=FALSE))

coef(mod, what="mu")
#> (Intercept)          x1 
#>   -2.421042    3.494641 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>    1.095616   -2.143970 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>   0.9816411