Xun-Jian Li

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.

The minorization–maximization (MM) algorithm is an optimization technique for iteratively calculating the maximizer of a concave target function rather than a root–finding tool. In this paper, we in the first time develop the MM algorithm as a new method for seeking the root of a univariate nonlinear equation . The key idea is to transfer the root–finding issue to iteratively calculate the maximizer of a concave target function by designing a new MM algorithm. According to the ascent property of the MM algorithm, we know that the proposed algorithm converges to the root and does not depend on any initial values, in contrast to Newton's method. Several statistical examples are provided to demonstrate the proposed algorithm.

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.