The function DGEII() defines the Discrete generalized exponential distribution,
Second type, a two parameter
distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
DGEII(mu.link = "logit", sigma.link = "log")defines the mu.link, with "logit" link as the default for the mu parameter. Other links are "probit" and "cloglog"'(complementary log-log).
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 DGEII distribution
in the gamlss() function.
The DGEII distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and mass function given by
\(f(x | \mu, \sigma) = (1-\mu^{x+1})^{\sigma}-(1-\mu^x)^{\sigma}\)
with \(0 < \mu < 1\) and \(\sigma > 0\). If \(\sigma=1\), the DGEII distribution reduces to the geometric distribution with success probability \(1-\mu\).
Note: in this implementation we changed the original parameters \(p\) to \(\mu\) and \(\alpha\) to \(\sigma\), we did it to implement this distribution within gamlss framework.
Nekoukhou V, Alamatsaz MH, Bidram H (2013). “Discrete generalized exponential distribution of a second type.” Statistics, 47(4), 876-887.
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(189)
y <- rDGEII(n=100, mu=0.75, sigma=0.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=DGEII,
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)
#> 0.6624943
exp(coef(mod1, what="sigma"))
#> (Intercept)
#> 0.7371248
# 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 <- rDGEII(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(1234)
datos <- gendat(n=100)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=DGEII, data=datos,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family: c("DGEII", "Discrete generalized exponential distribution of a second type II" )
#>
#> Call: gamlss(formula = y ~ x1, sigma.formula = ~x2, family = DGEII,
#> 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) 1.3850 0.1247 11.10 < 2e-16 ***
#> x1 -2.3913 0.2807 -8.52 1.95e-13 ***
#> ---
#> 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) 0.9022 0.2130 4.235 5.16e-05 ***
#> x2 1.1367 0.3155 3.603 0.000496 ***
#> ---
#> 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: 15
#>
#> Global Deviance: 424.2498
#> AIC: 432.2498
#> SBC: 442.6704
#> ******************************************************************
# Example 3
# Number of accidents to 647 women working on H. E. Shells
# for 5 weeks. Taken from
# Nekoukhou V, Alamatsaz MH, Bidram H (2013) page 886.
y <- rep(x=0:5, times=c(447, 132, 42, 21, 3, 2))
mod3 <- gamlss(y~1, family=DGEII,
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(mod3, what="mu"))
#> (Intercept)
#> 0.3378512
exp(coef(mod3, what="sigma"))
#> (Intercept)
#> 0.8985223