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. and Akantziliotou, C. (2006) Modelling the parameters of a family of mixed Poisson distributions including the Sichel and Delaptorte. Submitted for publication.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2003) 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.

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 5.0-2 Index]