The function HYPERPO() defines the hyper Poisson distribution, a two parameter
distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
HYPERPO(mu.link = "log", sigma.link = "log")defines the mu.link, with "log" link as the default for the mu parameter.
defines the sigma.link, with "log" link as the default for the sigma.
Returns a gamlss.family object which can be used
to fit a hyper-Poisson distribution
in the gamlss() function.
The hyper-Poisson distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and density given by
\(f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}\)
where the function \(_1F_1(a;c;z)\) is defined as
\(_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}\)
and \((a)_r = \frac{\gamma(a+r)}{\gamma(a)}\) for \(a>0\) and \(r\) positive integer.
Note: in this implementation we changed the original parameters \(\lambda\) and \(\gamma\) for \(\mu\) and \(\sigma\) respectively, we did it to implement this distribution within gamlss framework.
Sáez-Castillo AJ, Conde-Sánchez A (2013). “A hyper-Poisson regression model for overdispersed and underdispersed count data.” Computational Statistics & Data Analysis, 61, 148--157.
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rHYPERPO(n=200, mu=10, sigma=1.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=HYPERPO,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
#> (Intercept)
#> 9.66078
exp(coef(mod1, what="sigma"))
#> (Intercept)
#> 1.281153
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ HYPERPO
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1.21 - 3 * x1) # 0.75 approximately
sigma <- exp(1.26 - 2 * x2) # 1.30 approximately
y <- rHYPERPO(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(1235)
datos <- gendat(n=150)
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=HYPERPO, data=datos,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family: c("HYPERPO", "Hyper-Poisson")
#>
#> Call: gamlss(formula = y ~ x1, sigma.formula = ~x2, family = HYPERPO,
#> data = datos, control = gamlss.control(n.cyc = 500, trace = FALSE))
#>
#> Fitting method: RS()
#>
#> ------------------------------------------------------------------
#> Mu link function: log
#> Mu Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.0859 0.1554 6.988 8.82e-11 ***
#> x1 -3.6229 0.4252 -8.520 1.68e-14 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> Sigma link function: log
#> Sigma Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.7293 0.3288 2.218 0.028065 *
#> x2 -2.1551 0.5807 -3.711 0.000291 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> No. of observations in the fit: 150
#> Degrees of Freedom for the fit: 4
#> Residual Deg. of Freedom: 146
#> at cycle: 66
#>
#> Global Deviance: 301.8221
#> AIC: 309.8221
#> SBC: 321.8646
#> ******************************************************************