The Quasi XGamma Poisson family
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
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.
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)
#> 290251
exp(coef(mod, what='sigma'))
#> (Intercept)
#> 1.641354
exp(coef(mod, what='nu'))
#> (Intercept)
#> 2.499378
# 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))
#> Error in gamlss(x ~ x1, sigma.fo = ~x2, nu.fo = ~1, family = QXGP, control = gamlss.control(n.cyc = 5000, trace = FALSE)): response variable out of range
coef(mod, what="mu")
#> (Intercept)
#> 12.5785
coef(mod, what="sigma")
#> (Intercept)
#> 0.4955212
exp(coef(mod, what="nu"))
#> (Intercept)
#> 2.499378