The Tempered Discrete Linnik distribution

Abstract

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.

Appendix

Result 1. We provide a result on the p.f. of a family of r.v.’s displaying a very general type of p.g.f., which encompasses (13)—and hence (3), (10), (16) and (19)—as special cases. Let us consider an integer-valued r.v. X with p.g.f. given by

$$\begin{aligned} g_X(s)=\varphi (\alpha +\beta (1-\phi s)^\gamma ), \ s\in [0,1], \end{aligned}$$

where $\alpha ,\beta ,\gamma ,\phi \in \mathbb {R}$ are parameters in such a way that $\phi \in [0,1]$, while $\varphi :\mathbb {R}\mapsto [0,1]$ is a suitable function. As an example, the p.g.f. (19) of the r.v. $X_{TDL}$ is obtained by setting $\alpha =1-\mathrm{sgn}(a)bd(1-c)^a$, $\beta =\mathrm{sgn}(a)bd$, $\phi =c$ and $\gamma =a$ and by assuming that $\varphi (x)=x^{-1/d}$. As a further example, the p.g.f. (10) of the r.v. $X_{TDS}$ is achieved by setting $\alpha =\mathrm{sgn}(a)b(1-c)^a$, $\beta =-\mathrm{sgn}(a)b$, $\phi =c$ and $\gamma =a$, while $\varphi (x)=\exp (x)$. If the function $\varphi$ is analytic in a neighbourhood of $(\alpha +\beta )$, for $k\in \mathbb {N}$ the p.f. $p_X$ of the r.v. X may be expressed as

$$\begin{aligned} \begin{aligned} p_X(k)&=\frac{1}{k!}\,\left. \frac{d^kg_X(s)}{ds^k}\right| _{s=0}=\frac{1}{k!} \left. \frac{d^k\varphi (\alpha +\beta +\beta ((1-\phi s)^\gamma -1)}{ds^k}\,\right| _{s=0}\\&=\frac{1}{k!}\,\sum _{m=0}^k\frac{\beta ^m}{m!}\,\left. \frac{d^m \varphi (s)}{ds^m}\right| _{s=\alpha +\beta }\,\left. \frac{d^k((1-\phi s)^\gamma -1)^m}{ds^k}\right| _{s=0} \end{aligned} \end{aligned}$$

and, by means of the Binomial Theorem, it follows that

$$\begin{aligned} \begin{aligned} p_X(k)&=\frac{1}{k!}\,\sum _{m=0}^k\frac{\beta ^m}{m!}\left. \frac{d^m \varphi (s)}{ds^m}\right| _{s=\alpha +\beta }\,\sum _{j=0}^m(-1)^{m-j}\left( {\begin{array}{c}m\\ j\end{array}}\right) \left. \frac{d^k(1-\phi s)^{\gamma j}}{ds^k}\right| _{s=0}\\&=\frac{(-\phi )^k}{k!}\,\sum _{m=0}^k\beta ^m\,\left. \frac{d^m\varphi (s)}{ds^m} \right| _{s=\alpha +\beta }\,\frac{1}{m!}\,\sum _{j=0}^m(-1)^{m-j}\left( {\begin{array}{c}m\\ j\end{array}}\right) ( \gamma j)_k\\&=\frac{(-\phi )^k}{k!}\,\sum _{m=0}^k\beta ^m\,\left. \frac{d^m\varphi (s)}{ds^m} \right| _{s=\alpha +\beta }\,C(k,m,\gamma ), \end{aligned} \end{aligned}$$

where the generalized factorial coefficient and the falling factorial are defined in Sect. 4.

Result 2. On the basis of expression (19), it turns out that

$$\begin{aligned} g_{X_{TDL}}'(s)=(1+ \mathrm{sgn}(a)bd((1-cs)^a-(1-c)^a))^{-1}|a|bc(1-cs)^{a-1}g_{X_{TDL}}(s), \end{aligned}$$

from which

$$\begin{aligned} (K(1-cs)+(1-K)(1-cs)^{1-a})g_{X_{TDL}}'(s)=\frac{Kac}{d}\,g_{X_{TDL}}(s), \end{aligned}$$

(27)

where K is defined in Sect. 4. It is worth noting that

$$\begin{aligned} (1-cs)^{1-a}=1-\sum _{k=1}^\infty r_ks^k, \end{aligned}$$

where in turn the $r_k$’s are defined in Sect. 4. Hence, from (27) it follows that

$$\begin{aligned} \left( 1-Kcs-(1-K)\,\sum _{k=1}^\infty r_ks^k\right) \,\sum _{k=1}^\infty kp_{X_{TDL}}(k)s^{k-1}=\frac{Kac}{d}\,\sum _{k=0}^\infty p_{X_{TDL}}(k)s^k. \end{aligned}$$

(28)

Since

$$\begin{aligned}&\left( 1-Kcs-(1-K)\,\sum _{k=1}^\infty r_ks^k\right) \,\sum _{k=1}^\infty kp_{X_{TDL}}(k)s^{k-1}=\\&\sum _{k=0}^\infty \left( (k+1)p_{X_{TDL}}(k+1)-Kckp_{X_{TDL}}(k)-(1-K)\sum _{j=1}^kjr_{k-j+1}p_{X_{TDL}}(j)\right) s^k \end{aligned}$$

equating the coefficients of the powers of s in expression (28) the recursive relation (26) is obtained.