HYPERPO function

The hyper Poisson family

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.

Returns

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.

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)

References

\insertRef saez2013hyperpoDiscreteDists

See Also

dHYPERPO .

Author(s)

Freddy Hernandez, fhernanb@unal.edu.co