DPO: The Double Poisson distribution

Description

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.

Usage

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)

Arguments

mu.link

the link function for mu with default log

sigma.link

the link function for sigma with default log

x, q

vector of (non-negative integer) quantiles

vector of probabilities

the mu parameter

sigma

the sigma parameter

lower.tail

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)

max.value

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

how many random values to generate

Value

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

Details

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

$$f(y|\mu,\sigma)=\left( \frac{1}{\sigma} \right)^{1/2} e^{- \mu / \sigma} \left( \frac{e^{-y} y^{y}}{y!}\right) \left(\frac{e \mu}{y} \right)^{y/\sigma} C$$ for $y=0,1,2,\ldots,\infty$, $\mu>0$ and $\sigma>0$ where $C$ is the constant of proportinality which is calculated numerically using the function get_C.

References

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.

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.

Examples

Run this code

# NOT RUN {
DPO()
# 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)
plot(table(Y))
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)
# }

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