The hyper Poisson family — HYPERPO • DiscreteDists

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")

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 in the gamlss() function.

Details

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.

References

Sáez-Castillo, A. J., & Conde-Sánchez, A. (2013). A hyper-Poisson regression model for overdispersed and underdispersed count data. Computational Statistics & Data Analysis, 61, 148-157.

See also

Author

Freddy Hernandez, fhernanb@unal.edu.co

Examples

# 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=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 1024.801 

# 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(1234)
dat <- gendat(n=100)

mod2 <- gamlss(y~x1, sigma.fo=~x2, family=HYPERPO, data=dat,
                 control=gamlss.control(n.cyc=500, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 227.5195 
#> GAMLSS-RS iteration 2: Global Deviance = 226.6622 
#> GAMLSS-RS iteration 3: Global Deviance = 225.8377 
#> GAMLSS-RS iteration 4: Global Deviance = 225.021 
#> GAMLSS-RS iteration 5: Global Deviance = 224.2109 
#> GAMLSS-RS iteration 6: Global Deviance = 223.4426 
#> GAMLSS-RS iteration 7: Global Deviance = 222.7289 
#> GAMLSS-RS iteration 8: Global Deviance = 222.0889 
#> GAMLSS-RS iteration 9: Global Deviance = 221.5224 
#> GAMLSS-RS iteration 10: Global Deviance = 221.0354 
#> GAMLSS-RS iteration 11: Global Deviance = 220.629 
#> GAMLSS-RS iteration 12: Global Deviance = 220.293 
#> GAMLSS-RS iteration 13: Global Deviance = 220.0312 
#> GAMLSS-RS iteration 14: Global Deviance = 219.8322 
#> GAMLSS-RS iteration 15: Global Deviance = 219.6854 
#> GAMLSS-RS iteration 16: Global Deviance = 219.5778 
#> GAMLSS-RS iteration 17: Global Deviance = 219.5003 
#> GAMLSS-RS iteration 18: Global Deviance = 219.447 
#> GAMLSS-RS iteration 19: Global Deviance = 219.409 
#> GAMLSS-RS iteration 20: Global Deviance = 219.3824 
#> GAMLSS-RS iteration 21: Global Deviance = 219.3638 
#> GAMLSS-RS iteration 22: Global Deviance = 219.3511 
#> GAMLSS-RS iteration 23: Global Deviance = 219.3423 
#> GAMLSS-RS iteration 24: Global Deviance = 219.3364 
#> GAMLSS-RS iteration 25: Global Deviance = 219.3324 
#> GAMLSS-RS iteration 26: Global Deviance = 219.3296 
#> GAMLSS-RS iteration 27: Global Deviance = 219.3276 
#> GAMLSS-RS iteration 28: Global Deviance = 219.3263 
#> GAMLSS-RS iteration 29: Global Deviance = 219.3255 

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 = dat, control = gamlss.control(n.cyc = 500, trace = TRUE)) 
#> 
#> Fitting method: RS() 
#> 
#> ------------------------------------------------------------------
#> Mu link function:  log
#> Mu Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   0.8193     0.1822   4.497 1.90e-05 ***
#> x1           -3.0455     0.5279  -5.769 9.32e-08 ***
#> ---
#> 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)   0.8514     0.3645   2.336 0.021532 *  
#> x2           -2.5333     0.6995  -3.622 0.000466 ***
#> ---
#> 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:  29 
#>  
#> Global Deviance:     219.3255 
#>             AIC:     227.3255 
#>             SBC:     237.7461 
#> ******************************************************************