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Score matching is an estimation procedure that has been developed for
statistical models whose probability density function is known up to
proportionality but whose normalizing constant is intractable. For such models,
maximum likelihood estimation will be difficult or impossible to implement. To
date, nearly all applications of score matching have focused on continuous IID
(independent and identically distributed) models. Motivated by various data
modelling problems for which the continuity assumption and/or the IID
assumption are not appropriate, this article proposes three novel extensions of
score matching: (i) to univariate and multivariate ordinal data (including
count data); (ii) to INID (independent but not necessarily identically
distributed) data models, including regression models with either a continuous
or a discrete ordinal response; and (iii) to a class of dependent data models
known as auto models. Under the INID assumption, a unified asymptotic approach
to settings (i) and (ii) is developed and, under mild regularity conditions, it
is proved that the proposed score matching estimators are consistent and
asymptotically normal. These theoretical results provide a sound basis for
score-matching-based inference and are supported by strong performance in
simulation studies and a real data example involving doctoral publication data.
Regarding (iii), motivated by a spatial geochemical dataset, we develop a novel
auto model for spatially dependent spherical data and propose a
score-matching-based Wald statistic to test for the presence of spatial
dependence. Our proposed auto model exhibits a way to model spatial dependence
of directions, is computationally convenient to use and is expected to be
superior to composite likelihood approaches for reasons that are explained.