# dpoistweedie: The individual probabilities of Y when Y follows a... In poistweedie: Poisson-Tweedie exponential family models

## Description

Let X be a non-negative random variable following T. 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 individual probabilities of Y when Y follows a Poisson-Tweedie Distributions are: funtion Probabilitie.
For p = 1, it is a Neyman type A distribution; for p compris entre 1 et 2 , then Poisson-compound Poisson distribution is obtained; for p = 2,the Poisson-Tweedie model PTM correspond to the negative binomiale law lois BN; and, for p = 3, it is the Sichel or Poisson-inverse Gaussian distribution (e.g. Willmot, 1987). Also, when p tends vers l'infini, lamda and the mu egal to lambda and theta0 , the Poisson-Tweedie model PTM correspond to the poisson law Law of Poisson.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```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) ```

## Arguments

 `y` vector of (non-negative integer) quantiles y are the integer. `p` is a real index related to a precise model p. `n` non-negative integer (length of y) `x0` is a real index `mu` the meanmu, . `omega` is a real index.omega `lambda` the dispersion parameter lamda. `theta0` the canonical parameter theta0. `log` logical; if TRUE, probabilities y are given as log(y).

## Details

The Poisson-Tweedie distributions arethe EDMs with a variance of the form formule de la variance PT, where phi a generally implicit, denotes the inverse of the increansing function fonction inverse de W. omega(p,mu,theta0) is a function whose permit to determine the value of w.

## Value

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

## Author(s)

Cactha David Pechel, Laure Pauline Fotso and Celestin C Kokonendji Maintainer: Cactha David Pechel ( <[email protected]>)

## 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.

`ppoistweedie`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46``` ``` ## 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) ```