标题： 因子分析最大似然估计中的多重根现象 显示英文标题

标题： The Multiple Roots Phenomenon in Maximum Likelihood Estimation for Factor Analysis

摘要： 在统计推断中，多重根估计问题在文献中有许多不同的上下文。 在最大似然估计的背景下，多重根的存在会导致使用梯度上升算法计算最大似然估计量时出现不确定性，并进而导致统计推断中的困难。 在本文中，我们研究因子分析中最大似然估计的多重根现象。 我们证明，即使在最简单的情况下，相应的似然方程也有不可数多个可行解。 对于观测数据为二维且未观测因子得分是一维的情况，我们证明似然方程的解形成一个一维实曲线。 显示英文摘要

摘要： Multiple root estimation problems in statistical inference arise in many contexts in the literature. In the context of maximum likelihood estimation, the existence of multiple roots causes uncertainty in the computation of maximum likelihood estimators using hill-climbing algorithms, and consequent difficulties in the resulting statistical inference. In this paper, we study the multiple roots phenomenon in maximum likelihood estimation for factor analysis. We prove that the corresponding likelihood equations have uncountably many feasible solutions even in the simplest cases. For the case in which the observed data are two-dimensional and the unobserved factor scores are one-dimensional, we prove that the solutions to the likelihood equations form a one-dimensional real curve.