The GGEO family — GGEO • DiscreteDists

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")

Arguments

mu.link: defines the mu.link, with "log" link as the default for the mu parameter.
sigma.link: defines the sigma.link, with "logit" link as the default for the sigma. Other links are "probit" and "cloglog"'(complementary log-log)

Value

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

Details

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 \(1-\mu\).

References

Gómez-Déniz, E. (2010). Another generalization of the geometric distribution. Test, 19, 399-415.

See also

Author

Valentina Hurtado Sepúlveda, vhurtados@unal.edu.co

Examples

# 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 

# Example 4
# Total number of carious teeth among the four deciduous molars
# Taken from Gomez-Deniz (2010) page 412.

y <- rep(x=0:4, times=c(64, 17, 10, 6, 3))

mod4 <- gamlss(y~1, family=GGEO,
               control=gamlss.control(n.cyc=500, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 224.1384 

# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))

inv_logit(coef(mod4, what="mu"))
#> (Intercept) 
#>   0.4673054 
exp(coef(mod4, what="sigma"))
#> (Intercept) 
#>   0.6781431 

# Example 5
# Results of ten shots fired from a rifle at each of 100 targets.
# Taken from Gomez-Deniz (2010) page 412.

y <- rep(x=0:10, times=c(0, 2, 4, 10, 22, 26, 18, 12, 4, 2, 0))

mod5 <- gamlss(y~1, family=GGEO,
               control=gamlss.control(n.cyc=500, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 381.7361 

# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))

inv_logit(coef(mod5, what="mu"))
#> (Intercept) 
#>   0.3363266 
exp(coef(mod5, what="sigma"))
#> (Intercept) 
#>    394.4408 

# Example 6
# Fish catch data in Kemp (1992).
# Taken from Gomez-Deniz (2010) page 412.

y <- rep(x=0:9, times=c(1, 2, 11, 20, 29, 23, 10, 3, 1, 0))

mod6 <- gamlss(y~1, family=GGEO,
               control=gamlss.control(n.cyc=500, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 355.2735 

# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))

inv_logit(coef(mod6, what="mu"))
#> (Intercept) 
#>   0.2879751 
exp(coef(mod6, what="sigma"))
#> (Intercept) 
#>    283.5341