DPO: The Double Poisson distribution

DPOR Documentation

The Double Poisson distribution


The function DPO() defines the double Poisson distribution, a two parameters distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dDPO, pDPO, qDPO and rPO define the density, distribution function, quantile function and random generation for the double Poisson, DPO(), distribution. The function get_C() calculates numericaly the constant of proportionality needed for the pdf to sum up to 1.


DPO(mu.link = "log", sigma.link = "log")
dDPO(x, mu = 1, sigma = 1, log = FALSE)
pDPO(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qDPO(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE, 
     max.value = 10000)
rDPO(n, mu = 1, sigma = 1, max.value = 10000)
get_C(x, mu, sigma)



the link function for mu with default log


the link function for sigma with default log

x, q

vector of (non-negative integer) quantiles


vector of probabilities


the mu parameter


the sigma parameter


logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

log, log.p

logical; if TRUE, probabilities p are given as log(p)


a constant, set to the default value of 10000 for how far the algorithm should look for q


how many random values to generate


The definition for the Double Poisson distribution first introduced by Efron (1986) is:

f(y|mu,sigma)=(1/sigma)^{1/2} [(e^{-y} y^y)/y!] [(e mu)/y]^{y/sigma} C

for y=0,1,2, ...,Inf, μ>0 and σ>0 where C is the constant of proportinality which is calculated numerically using the function get_C.


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


The distributons calculates the constant of proportionality numerically therefore it can be slow for large data


Mikis Stasinopoulos, Bob Rigby and Marco Enea


Efron, B., 1986. Double exponential families and their use in generalized linear Regression. Journal of the American Statistical Association 81 (395), 709-721.

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, doi: 10.1201/9780429298547. 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, doi: 10.18637/jss.v023.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. doi: 10.1201/b21973

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

See Also



# overdisperse DPO
x <- 0:20
plot(x, dDPO(x, mu=5, sigma=3), type="h", col="red")
# underdisperse DPO
plot(x, dDPO(x, mu=5, sigma=.3), type="h", col="red")
# generate random sample
 Y <- rDPO(100,5,.5)
points(0:20, 100*dDPO(0:20, mu=5, sigma=.5)+0.2,  col="red")
# fit a model to the data 
# library(gamlss)
# gamlss(Y~1,family=DPO)

gamlss.dist documentation built on Aug. 28, 2022, 5:05 p.m.