Under-dispersed count data often appear in clinical trials, medical studies, demography, actuarial science, ecology, biology, industry and engineering. Although the generalized Poisson (GP) distribution possesses the twin properties of under- and over-dispersion, in the past 50 years, many authors only treat the GP distribution as an alternative to the negative binomial distribution for modeling over-dispersed count data. To our best knowledge, the issues of calculating maximum likelihood estimates (MLEs) of parameters in GP model without covariates and with covariates for the case of under-dispersion were not solved up to now. In this paper, we first develop a new minimization–maximization (MM) algorithm to calculate the MLEs of parameters in the GP distribution with under-dispersion, and then we develop another new MM algorithm to compute the MLEs of the vector of regression coefficients for the GP mean regression model for the case of under-dispersion. Three hypothesis tests (i.e., the likelihood ratio, Wald and score tests) are provided. Some simulations are conducted. The Bangladesh demographic and health surveys dataset is analyzed to illustrate the proposed methods and comparisons with the existing Conway–Maxwell–Poisson regression model are also presented.

In this paper, we propose a new and broadly applicable root-finding method, called as the upper-crossing/solution (US) algorithm, which belongs to the category of non-bracketing (or open domain) methods. The US algorithm is a general principle for iteratively seeking the unique root $\theta^{*}$ of a non-linear equation $g(\theta)=0$ and its each iteration consists of two steps: an upper-crossing step (U-step) and a solution step (S-step), where the U-step finds an upper-crossing function or a $U$-function $U(\theta|\theta^{(t)})$ [whose form depends on $\theta^{(t)}$ being the $t$-th iteration of $\theta^{*}$] based on a new notion of so-called changing direction inequality, and the S-step solves the simple $U$-equation $U(\theta|\theta^{(t)}) =0$ to obtain its explicit solution $\theta^{(t+1)}$. The US algorithm holds two major advantages: (i) It strongly stably converges to the root $\theta^{*}$; and (ii) it does not depend on any initial values, in contrast to Newton's method. The key step for applying the US algorithm is to construct one simple $U$-function $U(\theta|\theta^{(t)})$ such that an explicit solution to the $U$-equation $U(\theta|\theta^{(t)}) =0$ is available. Based on the first-, second- and third-derivative of $g(\theta)$, three methods are given for constructing such $U$-functions. We show various applications of the US algorithm in such as calculating quantile in continuous distributions, calculating exact $p$-values for skew null distributions, and finding maximum likelihood estimates of parameters in a class of continuous/discrete distributions. The analysis of the convergence rate of the US algorithm and some numerical experiments are also provided. Especially, because of the property of strongly stable convergence, the US algorithm could be one of the powerful tools for solving an equation with multiple roots.

The quadratic bound (QB) principle proposed by Böhning and Lindsay in 1988 is an important special case of the majorization–minimization or minorization-maximization optimization principle. The quadratic upper-bound (QUB) principle is pertinent to minimization; the analogous quadratic lower-bound principle is pertinent to maximization. Unfortunately, in minimizing a loss f ( θ ) , the QUB principle is limited by the difficulty of finding a constant positive definite matrix B such that B − d 2 f ( θ ) is positive semidefinite for all θ . This paper proposes a generalization of the QB principle that avoids this limitation. In particular, we construct QUB algorithms by replacing the matrix B by a continuous matrix-valued function B n ( θ ) that dominates the Hessian d 2 f ( θ ) and depends on the both the current iterate θ n and the next potential iterate θ . In practice, we require B n ( θ ) to be diagonal with its diagonal entries separated in θ . In other words, the i th diagonal entry of B n ( θ ) depends on θ only through its i th entry θ i . Theoretical analysis confirms that this class of generalized QB algorithms enjoys global convergence in favorable circumstances. For the scalar case, the tangency condition that the second derivative equals the bound function at the current point promotes superlinear convergence. Several numerical experiments implemented in Julia illustrate the power of the generalized QB principle.