## 1 Introduction  
Under-dispersed count data often appear in clinical trials, medical studies, demography, actuarial science, ecology, biology, industry and engineering. Examples include the number of embryonic deaths in mice in a clinical experiment [1], the number of power outages on each of 648 circuits in a power distribution system in the southeastern United States [2], the number of automotive services purchased on each visit for a customer at a US automotive services firm [3], the species richness that is the simplest measure of species diversity [4], the number of births during a period for women who live in Bangladesh ([[https://www.dhsprogram.com/data](https://www.dhsprogram.com/data)]([https://www.dhsprogram.com/data](https://www.dhsprogram.com/data)), accessed on 28 January 2022) and so on. The Poisson distribution is suitable for modeling equally dispersed count data, while the negative binomial distribution is often utilized to model over-dispersed count data. To fit under-dispersed count data, theoretically speaking, researchers should employ the _generalized Poisson_ (GP) distribution because it possesses the twin properties of under- and over-dispersion [5; 6; 7; 8; 9; 10; 11; 12; 13]. However, in the past 50 years, most authors just treat the GP distribution as an alternative to the negative binomial distribution by eyeing the former's over-dispersion property, while seeming to ignore its under-dispersion characteristic [6; 8; 9; 10; 11; 12; 13],although [PERSON] [6] proved that there exist unique MLEs of parameters for both over- and under-dispersion cases. The main reason for hindering researchers from using the GP distribution with under-dispersion is that calculating the _maximum likelihood estimates_ (MLEs) of parameters in GP models without/with covariates by a stable algorithm is not so easy. To our best knowledge, the issue of calculating MLEs of parameters in the GP model without/with covariates for the case of under-dispersion was not solved up to now; in other words, we may not obtain the correct MLEs of parameters by using the existing algorithms, see Section 5. A non-negative integer-valued _random variable_ (r.v.) \(X\) is said to follow a _generalized Poisson_ (GP) distribution with parameters \(\lambda>0\) and \(\psi\), denoted by \(X\sim\mathrm{GP}(\lambda,\psi)\), if its _probability mass function_ (pmf) is given by [5,14]  
\[p(x|\lambda,\psi)=\left\{ \begin{array}{ll}\dfrac{\lambda(\lambda+\psi x)^{x- 1}\mathrm{e}^{-\lambda-\psi x}}{x!},&x=0,1,\ldots,\infty,\\ 0,&x>r,\text{ when }\psi<0,\end{array} \right.\]  
where \(\max(-1,-\lambda/r)<\psi<1\) and \(r\)\((\geqslant 4)\) is the largest positive integer for which \(\lambda+\psi r>0\) when \(\psi<0\). The expectation and variance of \(X\) are given by [14]  
\[E(X)=\frac{\lambda}{1-\psi}\quad\text{and}\quad\mathrm{Var}(X)=\frac{\lambda} {(1-\psi)^{3}},\]  
respectively. The \(\mathrm{GP}(\lambda,\psi)\) distribution reduces to the Poisson\((\lambda)\) when \(\psi=0\), and it has the twin properties of over-dispersion when \(\psi>0\) and under-dispersion when \(\psi<0\). To formulate the mean regression of the GP distribution, Consul & Famoye [7] introduced a so-called _Type I generalized Poisson_ (\(\mathrm{GP}^{(\mathrm{I})}\)) distribution, denoted by \(Y\sim\mathrm{GP}^{(\mathrm{I})}(\mu,\alpha)\), through the following reparameterizations:  
\[\mu=\lambda(1-\psi)^{-1}>0\quad\text{and}\quad\alpha=(1-\psi)^{-1}. \tag{1}\]  
It is easy to show that the pmf of \(Y\) is  
\[p(y|\mu,\alpha)=\left\{ \begin{array}{ll}\dfrac{\mu[\mu+(\alpha-1)y]^{y-1} \exp\{-[\mu+(\alpha-1)y]/\alpha\}}{\alpha^{y}y!},&y=0,1,\ldots,\infty,\\ 0,&y>m,\text{ when }\alpha<1,\end{array} \right. \tag{2}\]  
where \(\mu>0\), \(\alpha\geqslant\max(1/2,1-\mu/m)\) and \(m\)\((\geqslant 4)\) is the largest positive integer for which \(\mu+(\alpha-1)m>0\) when \(\alpha<1\). The mean and variance of \(Y\) are given by:  
\[E(Y)=\mu\quad\text{and}\quad\mathrm{Var}(Y)=\alpha^{2}\mu,\]  
respectively, where \(\alpha\) denotes the square root of the index of dispersion.