HYPERPO: The hyper Poisson family

In fhernanb/prueba: Este Paquete Es Para Hacer Mis Pruebas

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")

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.

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.

fhernanb/prueba documentation built on March 29, 2025, 6:42 p.m.