Graphical and sparse (inverse) covariance models have found widespread use in modern sample-starved high dimensional applications. A part of their wide appeal stems from the significantly low sample sizes required for the existence of estimators, especially in comparison with the classical full covariance model. For undirected Gaussian graphical models, the minimum sample size required for the existence of maximum likelihood estimators had been an open question for almost half a century, and has been recently settled. The very same question for pseudo-likelihood estimators has remained unsolved ever since their introduction in the '70s. Pseudo-likelihood estimators have recently received renewed attention as they impose fewer restrictive assumptions and have better computational tractability, improved statistical performance, and appropriateness in modern high dimensional applications, thus renewing interest in this longstanding problem. In this paper, we undertake a comprehensive study of this open problem within the context of the two classes of pseudo-likelihood methods proposed in the literature. We provide a precise answer to this question for both pseudo-likelihood approaches and relate the corresponding solutions to their Gaussian counterpart.

翻译：图形化与稀疏（逆）协方差模型在现代样本匮乏的高维应用中得到了广泛使用。其吸引力部分源于所需的最小样本量显著较低，这尤其体现在与经典全协方差模型的比较中。对于无向高斯图模型，最大似然估计量存在所需的最小样本量问题曾是近半个世纪以来的未解难题，而这一问题近期已获解决。然而，自20世纪70年代伪似然估计量提出以来，其相同的基本问题始终悬而未决。近年来，伪似然估计量因其假设限制更少、计算可处理性更优、统计性能改进及对现代高维应用的适用性而重新受到关注，从而激发了学界对这一长期问题的兴趣。本文针对文献中提出的两类伪似然方法，对该开放问题展开全面研究。我们为两种伪似然方法提供了该问题的精确回答，并将其对应解与高斯情形下的解建立了关联。