DiscreteDists (version 1.0.0)

HYPERPO: The hyper Poisson family

Description

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().

Usage

HYPERPO(mu.link = "log", sigma.link = "log")

Value

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

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.

Author

Freddy Hernandez,

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

saez2013hyperpoDiscreteDists

See Also

dHYPERPO.

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=FALSE))

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))

# 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(1235)
datos <- gendat(n=150)

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

summary(mod2)