标题： 双变量缺失数据最大似然估计的代数复杂性 显示英文标题

标题： The Algebraic Complexity of Maximum Likelihood Estimation for Bivariate Missing Data

摘要： 我们研究了在“数据缺失随机”（MAR）假设下，正态分布和多项分布随机变量的二元缺失数据模式下的最大似然估计问题。 对于正态数据，得分方程有九个复数解，其中至少有一个实数解且具有统计显著性。 我们的计算表明，实数解的数量与MAR假设是否成立有关。 在多项分布情况下，得分方程的所有解均为实数解，并且实数解的数量随着潜在随机变量状态数量的增长呈指数增长，但总是恰好存在一个具有统计显著性的局部极大值。 显示英文摘要

摘要： We study the problem of maximum likelihood estimation for general patterns of bivariate missing data for normal and multinomial random variables, under the assumption that the data is missing at random (MAR). For normal data, the score equations have nine complex solutions, at least one of which is real and statistically significant. Our computations suggest that the number of real solutions is related to whether or not the MAR assumption is satisfied. In the multinomial case, all solutions to the score equations are real and the number of real solutions grows exponentially in the number of states of the underlying random variables, though there is always precisely one statistically significant local maxima.