R: The generalised Poisson distribution

GPO {gamlss.dist} | R Documentation

## The generalised Poisson distribution

### Description

The `GPO()` function defines the generalised Poisson distribution, a two parameter discrete distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dGPO`, `pGPO`, `qGPO` and `rGPO` define the density, distribution function, quantile function and random generation for the Delaporte `GPO()`, distribution.

### Usage

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

    dGPO(x, mu = 1, sigma = 1, log = FALSE)

    pGPO(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

    qGPO(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE, max.value = 10000)

    rGPO(n, mu = 1, sigma = 1,  max.value = 10000)

### 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` parameter
`x` | vector of (non-negative integer) quantiles
`mu` | vector of positive `mu`
`sigma` | vector of positive dispersion parameter `sigma`
`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]
`max.value` | a constant, set to the default value of 10000 for how far the algorithm should look for q

### Details

The probability function of the Generalised Poisson distribution is given by

f(y|mu,sigma)= (μ/ (1+σ+μ))^y (1+σ y)^(y-1)/y! exp(-μ (1+σ y)/(1+σ+μ))

for y=0,1,2,... where mu>0 and σ>0.

### Value

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

### Author(s)

Rigby, R. A., Stasinopoulos D. M.

### 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.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

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, https://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.

(see also https://www.gamlss.com/).

### See Also

`gamlss.family`, `PO` , `DPO`

### Examples

    GPO()# gives information about the default links for the
    #plot the pdf using plot
    plot(function(y) dGPO(y, mu=10, sigma=1 ), from=0, to=100, n=100+1, type="h") # pdf
    # plot the cdf
    plot(seq(from=0,to=100),pGPO(seq(from=0,to=100), mu=10, sigma=1), type="h")   # cdf
    # generate random sample
    tN <- table(Ni <- rGPO(100, mu=5, sigma=1))
    r <- barplot(tN, col='lightblue')

[Package gamlss.dist version 5.3-2 Index]