On the differences of two generalized negative binomial variates

The generalized negative binomial (GNB) distribution was defined by Jain and Consul (SIAM J. Appl. Math., 21 (1971)) and was obtained as a particular family of Lagrangian distributions by Consul and Shenton (SIAM J. Appl. Math., 23 (1973)). Consul and Shenton also gave the probability generating function (p.g.f.) and proved many properties of the GNBD. Consul and Gupta (SIAM J. Appl. Math., 39 (1980)) proved that the parameter β must be either zero or 1≤ β ≤ θ^-1 for the GNBD to be a true probability distribution and proved some other properties. Numerous applications and properties of this model have been studied by various researchers. Considering two independent GNB variates X and Y, with parameters (m,β,θ) and (n,β,θ) respectively, the probability distribuition of D = Y-X and its p.g.f. and cumulant generating function have been obtained. A recurrence relation between the cumulants has been established and the first four cumulants, β₁ and β₂ have been derived. Also some moments of the absolute difference |Y-X| have been obtained.