The function GGEO() defines the Generalized Geometric distribution,
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
GGEO(mu.link = "logit", sigma.link = "log")Returns a gamlss.family object which can be used
to fit a GGEO distribution
in the gamlss() function.
The GGEO distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and mass function given by
\(f(x | \mu, \sigma) = \frac{\sigma \mu^x (1-\mu)}{(1-(1-\sigma) \mu^{x+1})(1-(1-\sigma) \mu^{x})}\)
with \(0 < \mu < 1\) and \(\sigma > 0\). If \(\sigma=1\), the GGEO distribution reduces to the geometric distribution with success probability \(1-\mu\).
gomez2010DiscreteDists
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123)
y <- rGGEO(n=200, mu=0.95, sigma=1.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=GGEO,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod1, what="mu"))
#> (Intercept)
#> 0.9433538
exp(coef(mod1, what="sigma"))
#> (Intercept)
#> 1.924394
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ GGEO
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- inv_logit(1.7 - 2.8*x1)
sigma <- exp(0.73 + 1*x2)
y <- rGGEO(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(78353)
datos <- gendat(n=100)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=GGEO, data=datos,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family: c("GGEO", "Generalized geometric")
#>
#> Call: gamlss(formula = y ~ x1, sigma.formula = ~x2, family = GGEO,
#> data = datos, control = gamlss.control(n.cyc = 500, trace = FALSE))
#>
#> Fitting method: RS()
#>
#> ------------------------------------------------------------------
#> Mu link function: logit
#> Mu Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.5645 0.1602 9.766 3.89e-16 ***
#> x1 -3.0030 0.3780 -7.945 3.32e-12 ***
#> ---
#> 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) 1.7825 0.3690 4.831 5.03e-06 ***
#> x2 -0.4900 0.6196 -0.791 0.431
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> No. of observations in the fit: 100
#> Degrees of Freedom for the fit: 4
#> Residual Deg. of Freedom: 96
#> at cycle: 19
#>
#> Global Deviance: 413.8106
#> AIC: 421.8106
#> SBC: 432.2313
#> ******************************************************************
# Example 3
# Number of accidents to 647 women working on H. E. Shells
# for 5 weeks. Taken from Gomez-Deniz (2010) page 411.
y <- rep(x=0:5, times=c(447, 132, 42, 21, 3, 2))
mod3 <- gamlss(y~1, family=GGEO,
control=gamlss.control(n.cyc=500, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 1184.469
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod3, what="mu"))
#> (Intercept)
#> 0.3399179
exp(coef(mod3, what="sigma"))
#> (Intercept)
#> 0.8738869