# The individual probabilities of Y when Y follows a Poisson-Tweedie Distributions 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 $Law of Poisson$. ```r 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 mean$mu$, . - `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). ## Returns density (`dpoistweedie`),for the given Poisson-Tweedie distribution with parameters ## 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. ## Author(s) Cactha David Pechel, Laure Pauline Fotso and Celestin C Kokonendji Maintainer: Cactha David Pechel ( ) ## 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 ```r ## 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) ```