gamlss.dist: DEL – R documentation

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DEL

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

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))

gamlss.dist

Distributions for Generalized Additive Models for Location Scale and Shape

v5.3-2

GPL-2 | GPL-3

Authors

Mikis Stasinopoulos [aut, cre, cph], Robert Rigby [aut], Calliope Akantziliotou [ctb], Vlasios Voudouris [ctb], Gillian Heller [ctb], Fernanda De Bastiani [ctb], Raydonal Ospina [ctb], Nicoletta Motpan [ctb], Fiona McElduff [ctb], Majid Djennad [ctb], Marco Enea [ctb], Alexios Ghalanos [ctb], Christos Argyropoulos [ctb], Almond Stocker [ctb], Jens Lichter [ctb], Stanislaus Stadlmann [ctb]

Initial release

2021-03-08

DEL