R: The Delaporte distribution for fitting a GAMLSS model

DEL {gamlss.dist} | R Documentation

## The Delaporte distribution for fitting a GAMLSS model

### Description

The `DEL()` function defines the Delaporte distribution, a three parameter discrete distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dDEL`, `pDEL`, `qDEL` and `rDEL` define the density, distribution function, quantile function and random generation for the Delaporte `DEL()`, distribution.

### Usage

    DEL(mu.link = "log", sigma.link = "log", nu.link = "logit")
    dDEL(x, mu=1, sigma=1, nu=0.5, log=FALSE)
    pDEL(q, mu=1, sigma=1, nu=0.5, lower.tail = TRUE,
            log.p = FALSE)
    qDEL(p, mu=1, sigma=1, nu=0.5,  lower.tail = TRUE,
         log.p = FALSE,  max.value = 10000)
    rDEL(n, mu=1, sigma=1, nu=0.5, 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
`nu.link` | Defines the `nu.link`, with "logit" link as the default for the nu parameter
`x` | vector of (non-negative integer) quantiles
`mu` | vector of positive mu
`sigma` | vector of positive dispersion parameter
`nu` | vector of nu
`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 Delaporte distribution is given by

f(y|mu,sigma,nu)=(exp(-mu*nu)/Gamma(1/sigma))*[1+mu*sigma*(1-nu)]^(-1/sigma) S

where

S=Sum(Per(y,j))*((mu^y)*(nu^{y-j})/y!) *[1+(1/(sigma*(1-nu)))]^j Gamma((1/sigma)*j)

for y=0,1,2,... where mu>0 , σ>0 and 0<nu<1. This distribution is a parametrization of the distribution given by Wimmer and Altmann (1999) p 515-516 where a=mu*nu, 1/sigma and p=[1+mu*nu*(1-nu)]^(-1)

### Value

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

### Note

The mean of Y is given by E(Y)=mu and the variance by V(Y)=mu+mu^2*sigma*(1-nu)^2.

### Author(s)

Rigby, R. A., Stasinopoulos D. M. and Marco Enea

### 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.

Wimmer, G. and Altmann, G (1999). Thesaurus of univariate discrete probability distributions . Stamn Verlag, Essen, Germany

### See Also

`gamlss.family`, `SI` , `SICHEL`

### Examples

     DEL()# gives information about the default links for the  Delaporte distribution
    #plot the pdf using plot
    plot(function(y) dDEL(y, mu=10, sigma=1, nu=.5), from=0, to=100, n=100+1, type="h") # pdf
    # plot the cdf
    plot(seq(from=0,to=100),pDEL(seq(from=0,to=100), mu=10, sigma=1, nu=0.5), type="h")   # cdf
    # generate random sample
    tN <- table(Ni <- rDEL(100, mu=10, sigma=1, nu=0.5))
    r <- barplot(tN, col='lightblue')
    # fit a model to the data
    # libary(gamlss)
    # gamlss(Ni~1,family=DEL, control=gamlss.control(n.cyc=50))

[Package gamlss.dist version 5.3-2 Index]