The Tempered Discrete Linnik distribution

We introduce a new family of integer-valued distributions by considering a tempered version of the Discrete Linnik law. The proposal is actually a generalization of the well-known Poisson–Tweedie law. The suggested family is extremely flexible since it contains a wide spectrum of distributions ranging from light-tailed laws (such as the Binomial) to heavy-tailed laws (such as the Discrete Linnik). The main theoretical features of the Tempered Discrete Linnik distribution are explored by providing a series of identities in law, which describe its genesis in terms of mixture Poisson distribution and compound Negative Binomial distribution—as well as in terms of mixture Poisson–Tweedie distribution. Moreover, we give a manageable expression and a suitable recursive relationship for the corresponding probability function. Finally, an application to scientometric data—which deals with the scientific output of the researchers of the University of Siena—is considered.