The hyper Poisson family (with mu as mean)

The function HYPERPO2() defines the hyper Poisson distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

HYPERPO2(mu.link = "log", 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 "log" link as the default for the sigma.

Value

Returns a gamlss.family object which can be used to fit a hyper-Poisson distribution version 2 in the gamlss() function.

Details

The hyper-Poisson distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ...

Note: in this implementation the parameter \(\mu\) is the mean of the distribution and \(\sigma\) corresponds to the dispersion parameter. If you fit a model with this parameterization, the time will increase because an internal procedure to convert \(\mu\) to \(\lambda\) parameter.

References

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.

Author

Freddy Hernandez, fhernanb@unal.edu.co

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rHYPERPO2(n=200, mu=3, sigma=0.5)

# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=HYPERPO2,
               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) 
#>    2.949987 
exp(coef(mod1, what='sigma'))
#> (Intercept) 
#>   0.4793693 

# Example 2
# Generating random values under some model

# \donttest{
# A function to simulate a data set with Y ~ HYPERPO2
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 <- rHYPERPO2(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(1234)
datos <- gendat(n=500)

mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=HYPERPO2, data=datos,
               control=gamlss.control(n.cyc=500, trace=FALSE))

summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family:  c("HYPERPO2", "Hyper-Poisson-2") 
#> 
#> Call:  gamlss(formula = y ~ x1, sigma.formula = ~x2, family = HYPERPO2,  
#>     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)  1.24765    0.07741   16.12   <2e-16 ***
#> x1          -3.26816    0.20479  -15.96   <2e-16 ***
#> ---
#> 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)   1.3782     0.5316   2.592  0.00981 **
#> x2           -2.4371     0.8745  -2.787  0.00553 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> ------------------------------------------------------------------
#> No. of observations in the fit:  500 
#> Degrees of Freedom for the fit:  4
#>       Residual Deg. of Freedom:  496 
#>                       at cycle:  3 
#>  
#> Global Deviance:     1119.295 
#>             AIC:     1127.295 
#>             SBC:     1144.153 
#> ******************************************************************
# }