# PO: Poisson distribution for fitting a GAMLSS model In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

This function `PO` defines the Poisson distribution, an one parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dPO`, `pPO`, `qPO` and `rPO` define the density, distribution function, quantile function and random generation for the Poisson, `PO()`, distribution.

## Usage

 ```1 2 3 4 5``` ```PO(mu.link = "log") dPO(x, mu = 1, log = FALSE) pPO(q, mu = 1, lower.tail = TRUE, log.p = FALSE) qPO(p, mu = 1, lower.tail = TRUE, log.p = FALSE) rPO(n, mu = 1) ```

## Arguments

 `mu.link` Defines the `mu.link`, with "log" link as the default for the mu parameter `x` vector of (non-negative integer) quantiles `mu` vector of positive means `p` vector of probabilities `q` vector of quantiles `n` number of random values to return `log, log.p` logical; if TRUE, probabilities p are given as log(p) `lower.tail` logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

## Details

Definition file for Poisson distribution.

f(y|μ)=e^(-μ)*μ^y/Γ(y+1)

for y=0,1,2,... and μ>0.

## Value

returns a `gamlss.family` object which can be used to fit a Poisson distribution in the `gamlss()` function.

## Note

mu is the mean of the Poisson distribution

## Author(s)

Bob Rigby, Mikis Stasinopoulos [email protected], and Kalliope Akantziliotou

## References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, http://www.jstatsoft.org/v23/i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.

`gamlss.family`, `NBI`, `NBII`, `SI`, `SICHEL`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```PO()# gives information about the default links for the Poisson distribution # fitting data using PO() # plotting the distribution plot(function(y) dPO(y, mu=10 ), from=0, to=20, n=20+1, type="h") # creating random variables and plot them tN <- table(Ni <- rPO(1000, mu=5)) r <- barplot(tN, col='lightblue') # library(gamlss) # data(aids) # h<-gamlss(y~cs(x,df=7)+qrt, family=PO, data=aids) # fits the constant+x+qrt model # plot(h) # pdf.plot(family=PO, mu=10, min=0, max=20, step=1) ```