  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.
