poistweedie (version 1.0.2)

dpoistweedie: The individual probabilities of Y when Y follows a Poisson-Tweedie Distributions

Description

Let X be a non-negative random variable following \(\mathcal{T}_{P}(\theta , \lambda)\). If a discrete random variable Y is such that the conditional distribution of Y given X is Poisson with mean X, then the EDM generated by the distribution of Y is of the Poisson-Tweedie class. For \(p>=1\) individual probabilities of \(Y\sim\mathcal{P\mathcal{T}_{P}(\theta ,\lambda )}\) when Y follows a Poisson-Tweedie Distributions are: \( Pr(Y=y)=\int_{0}^{\infty}\frac{e^{-x}x^{y}}{y!}\mathcal{T}_{P}(\theta , \lambda)d(x), y=0,1, \).
For p = 1, it is a Neyman type A distribution; for \(1<p<2\) , then Poisson-compound Poisson distribution is obtained; for p = 2,the Poisson-Tweedie model \(PT_{2}\left(\mu,\lambda\right) \) correspond to the negative binomiale law \(BN\left( \lambda,\frac{1}{1+\mu}\right)\); and, for p = 3, it is the Sichel or Poisson-inverse Gaussian distribution (e.g. Willmot, 1987). Also, when \(p\longrightarrow\infty\), \(\lambda=\frac{\mu\times\left( 1-\theta_{0}\right) }{1+\mu}\) and the \(\lambda=\mu\simeq-\theta_{0}\) , the Poisson-Tweedie model \(PT_{p}\left( \mu,\lambda\right)\) correspond to the poisson law \( P_{y}\left( \lambda^{2}\right)\).

Usage

dpoistweedie(y, p, mu, lambda, theta0, log)
       densitept1(p, n, mu, lambda, theta0)
       densitept2(p, n, mu, lambda, theta0)
       dpt1(p, n, mu, lambda, theta0)
       dpt1Log(p, n, mu, lambda, theta0)
       dpt2(p, n, mu, lambda, theta0)
       dpt2Log(p, n, mu, lambda, theta0)
       dptp(p, n, mu, lambda, theta0)
       dptpLog(p, n, mu, lambda, theta0)
       gam1.1(y, lambda)
       gam1.2(y, lambda)
       imfx0(x0,p,mu,theta0)
       moyennePT(p,omega,theta0)
       omega(p,mu,theta0)
       testOmegaPT(p,n)

Value

density (dpoistweedie),for the given Poisson-Tweedie distribution with parameters

Arguments

y

vector of (non-negative integer) quantiles \(Y=(y1,y2,...,yn) \ where \ \ yi i=1,2,...,n\) are the integer.

p

is a real index related to a precise model \(p>=1\).

n

non-negative integer (length of y)

x0

is a real index

mu

the mean\(\mu\in{R}^{+}\), .

omega

is a real index.\(\omega\in{R}\)

lambda

the dispersion parameter \(\lambda\in{R},\lambda>0\).

theta0

the canonical parameter \(\theta_{0}\in{R}^{-}\).

log

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

Author

Cactha David Pechel, Laure Pauline Fotso and Celestin C Kokonendji Maintainer: Cactha David Pechel ( <[e-mail]>)

Details

The Poisson-Tweedie distributions arethe EDMs with a variance of the form \(V_{p}^{\mathcal{PT}}\left( \mu\right) =\mu+\mu^{p}\exp\left\{ \left(2-p\right) \Phi_{p}\left( \mu\right) \right\} ,\mu>0\), where \(\Phi_{p}\left( \mu\right) \) a generally implicit, denotes the inverse of the increansing function \(\omega\longrightarrow\frac{d\left\{ \ln IE\left( e^{wy}\right) \right\}}{dw}\). omega(p,mu,theta0) is a function whose permit to determine the value of w.

References

Dunn, Peter K and Smyth, Gordon K (To appear). Series evaluation of Tweedie exponential dispersion model densities Statistics and Computing.

Dunn, Peter K and Smyth, Gordon K (2001). Tweedie family densities: methods of evaluation. Proceedings of the 16th International Workshop on Statistical Modelling, Odense, Denmark, 2--6 July

Hougaard, P., Lee, M-L.T. and Whitmore, G.A. (1997). Analysis of overdispersed count data by mixtures of Poisson variables and Poisson processes, Biometrics 53, 1225--1238

Jorgensen, B. (1987). Exponential dispersion models. Journal of the Royal Statistical Society, B, 49, 127--162.

Kokonendji, C.C., Demeetrio, C.G.B. and Dossou-Gbete, S. (2004). Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demetrio classes. SORT: Statistics and Operations Research Transactions 28 (2), 201--214.

See Also

ppoistweedie

Examples

## dpoistweedie(y, power, mu,lambda,theta0,log = FALSE)
## Plot dpois() and dpoistweedie() with  log=FALSE
layout(matrix(1 :1, 1, 1))
layout.show(2) 
power <- exp(10) 
mu <-10
lambda <- 10
theta0<--10
lambda1<-100
y <- 0:200
## plot  dpoistweedie function with log = FALSE
d1<-dpoistweedie(y,power,mu,lambda,theta0,log = FALSE)
d2<-dpois(y,lambda1,log=FALSE)
erreure<-d1-d2
plot (y,d1,col='blue', type='h',xlab="y 
   avec  y=0:200,  power=exp(30),mu=10, lambda=10,
   theta0=-10,  lambda1=100", ylab="densite P(100)",
   main = "dpoistweedie(*,col='blue' log=FALSE)
   et dpois(*,col='red' log=FALSE)")
lines(y,d2,type ="p",col='red',lwd=2)
sum(abs(erreure))



## Plot dnbinom() and dpoistweedie()
layout(matrix(1 :1, 1, 1))
layout.show(2) 
power<-2 
mu<-10
lambda <- 1
theta0<-0
prob<-1-(mu/(1+mu))
y <- seq(0,50, by =3)
## plot a dpoistweedie function with log=FALSE
d1<-dpoistweedie(y,power,mu,lambda,theta0,log=FALSE)
d2<-dnbinom(y,lambda,prob, log=FALSE)
erreure<-d1-d2
plot (y,d1,col='blue', type='h',xlab="y 
   avec  y=seq(0,50,by=3),  power=2,mu=10, 
   lambda=1, thetao=0", ylab="densite NB(1,1/11)"
   ,main = "dnpoistweedie(*,col='blue' log=FALSE)
   et dnbinom(*,col='red' log=FALSE)")
lines(y,d2,type ="p",col='red',lwd=2)
abs(erreure)