Some properties of a two-parameter family of Poisson distributions of order \(k\) (Q1286379)

Let \(S_j\), \(1\leq j\leq k\), be independent identically distributed random variables with Poisson distribution with parameter \(\lambda>0\). Then the distribution of the random variable \(S= S_1+ 2S_2+\cdots+ kS_k\) is called a Poisson distribution of order \(k\) [\textit{A. N. Philippou}, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 130, 175-180 (1983; Zbl 0529.60010)]. Suppose the random variable \(S\) is not directly observable but a random variable \(y\) is detected with probability \(\varepsilon\), \(0<\varepsilon \leq 1\), so that the conditional distribution of \(y\) given \(S\) is binomial with parameters \(S\) and \(\varepsilon\). Then the unconditional distribution of \(y\) has the probability generating function \(P_s(z)= \exp\{\lambda \sum_{j=1}^k [(1-\varepsilon +\varepsilon z)^j-1]\}\) and it gives rise to a two-parameter family of distributions for a given \(k\). If \(\varepsilon =1\), then it reduces to the Poisson distribution of order \(k\). Here the authors study properties of such distributions and develop recursion formulae for computing moments, semi-invariants, etc. and study asymptotics of the distribution as \(\lambda\to\infty\).