Linear Estimating Equations for Exponential Families with Application to Gaussian Linear Concentration Models

In many families of distributions, maximum likelihood estimation is intractable because the normalization constant for the density which enters into the likelihood function is difficult to compute. The score matching estimator of Hyv\"arinen (2005) provides an alternative where this normalization constant is not needed. The corresponding estimating equations become linear for an exponential family. The score matching estimator is shown to be consistent and asymptotically normally distributed for such models, although not necessarily efficient. Gaussian linear concentration models are an example of such a family; we show that the linear score matching equations of the unknown concentration matrix may be solved even when the number of variables is much larger than the number of observations. For linear concentration models that are also linear in the covariance we show that the score matching estimator is identical to the maximum likelihood estimator, hence in such cases it is also efficient. Gaussian graphical models with symmetries form a particularly interesting subclass of linear concentration models and we investigate the potential use of the score matching estimator for this case.