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This article describes the R package RNGforGPD, which is designed for the generation of univariate and multivariate generalized Poisson data. Some illustrative examples are given, the utility and functionality of the package are demonstrated; and its performance is assessed via simulations that are devised around both artificial and real data. Introduction and motivation It is well known that the variance of a Poisson variable equals to its mean. However, overand underdispersion in count data could make this mean-variance equality assumption unrealistically simplistic. This situation is often caused by the heterogeneity in the population, while we implicitly assume that the weights assigned to each event are equal when we employ the regular Poisson distribution. This problem can be addressed by modeling count data using the generalized Poisson distribution (GPD), which enables us to assign varying weights to events (Satterthwaite, 1942). The GPD includes a dispersion parameter λ, which accommodates overor under-dispersion relative to the Poisson distribution in addition to the rate parameter θ in the regular Poisson distribution. Scientific background As discussed in the book of Consul (1989), the GPD has two parameters, rate and dispersion. It can be regarded as a mixture of Poisson distributions according to Joe and Zhu (2005). The GPD can be described mathematically as follows: Let X be a discrete random variable defined over non-negative integers, and let Px(θ, λ) be its probability mass function (pmf). X is said to follow the GPD with rate parameter θ and dispersion parameter λ if Px(θ, λ) = { θ (θ + λx)x−1 e−θ−λx/x!, for x = 0, 1, 2... 0, for x > m if λ < 0 and zero otherwise, where θ > 0, max(−1,−θ/m) ≤ λ < 1, and m(≥ 4) is the largest positive integer for which θ + mλ > 0 when λ < 0. The parameters θ and λ are independent, but the lower limits on λ and m are imposed to ensure that there are at least five classes with non-zero probability when λ is negative. λ = 0 corresponds to the Poisson distribution, while λ > 0 and λ < 0 correspond to overand under-dispersion relative to the regular Poisson, respectively. Besides, if we regard the weights of each event in a time period as independent and formulate the summation of the weights as a characteristic function, the distribution function of the sum of weights has all the properties of the GPD after the application of the Fourier transformation (Satterthwaite, 1942). According to Vernic (2000), the pmf for the multivariate GPD can be derived using the multivariate reduction method. In her derivation, an m-dimensional GPD (MGP) is obtained by taking (m + 1) independent univariate generalized Poisson random variables, Xi ∼ GPD(θi, λi), for i = 0, ..., m, and let Y1 = X1 + X0, Y2 = X2 + X0, ..., Ym = Xm + X0, Then (Y1, ..., Ym) ∼ MGP(Θ, Λ), where Θ = (θ0, ..., θm) and Λ = (λ1, ..., λm). The joint pmf of (Y1, ..., Ym) is P(y1, ..., ym) = P(Y1 = y1, ..., Ym = ym) = min(y1,...,ym) ∑ k=0 p1(y1 − k) · ... · pm(ym − k)p0(k), where pi is the pmf of the random variable Xi. Plugging in the pmf of Xi’s, we get The R Journal Vol. XX/YY, AAAA 20ZZ ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLE 2 P(y1, ..., ym) =  m ∏ j=0 θj  exp −θ − m ∑ j=1 yjλj  · min(y1,...,ym) ∑ k=0  m ∏ j=1 [θj + (yj − k)λj]j (yj − k)!  · · (θ0 + kλ0) k−1 k! exp k  m ∑ j=1 λj − λ0  , where θ = ∑j=0 θj and 0! = 1.

RNGforGPD: An R Package for Generation of Univariate and Multivariate Generalized Poisson Data