Guo-Liang Tian | Searcharxiv

arXiv2026

The normalized expectation-maximization (N-EM) algorithm

Although the $\textit{expectation-maximization}$ (EM) algorithm is a powerful optimization tool in statistics, it can only be applied to missing/incomplete data problems or to problems with a latent-variable structure. It is well known that the introduction of latent variables (or the data augmentation) is an art; i.e., it could only be done case by case. In this paper, we propose a new algorithm, a so-called $\textit{normalized EM}$ (N-EM) algorithm, for a class of log-likelihood functions with integrals. As an extension of the original EM algorithm, the N-EM algorithm inherits all advantages of EM-type algorithms and consists of three steps: normalization step (N-step), expectation step (E-step) and maximization step (M-step), where the N-step is to construct a $\textit{normalized density function}$ (ndf), the E-step is to compute a well-established surrogate $Q$-function and the M-step is to maximize the $Q$-function as in the original EM algorithm. The ascent property, the best choice of the ndf, and those N-EM algorithms with a difficult M-step are also explored. By multiple real applications, we have shown that the N-EM algorithm can solve some problems which cannot be addressed by the EM algorithm. Next, for problems to which the EM can be applied (often case by case), the N-EM algorithm can be employed in a unified framework. Numerical experiments are performed and convergence properties are also established.

stat.ME↗

arXiv2025

The upper-crossing/solution (US) algorithm for root-finding with strongly stable convergence

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 $θ^{*}$ of a non-linear equation $g(θ)=0$ 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 $U(θ|θ^{(t)})$ [whose form depends on $θ^{(t)}$ being the $t$-th iteration of $θ^{*}$] based on a new notion of so-called changing direction inequality, and the S-step solves the simple $U$-equation $U(θ|θ^{(t)}) =0$ to obtain its explicit solution $θ^{(t+1)}$. The US algorithm holds two major advantages: (i) It strongly stably converges to the root $θ^{*}$; 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 $U(θ|θ^{(t)})$ such that an explicit solution to the $U$-equation $U(θ|θ^{(t)}) =0$ is available. Based on the first-, second- and third-derivative of $g(θ)$, 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.

math.NA↗