  Many probabilistic models that have an intractable normalizing constant may
be extended to contain covariates. Since the evaluation of the exact likelihood
is difficult or even impossible for these models, score matching was proposed
to avoid explicit computation of the normalizing constant. In the literature,
score matching has so far only been developed for models in which the
observations are independent and identically distributed (IID). However, the
IID assumption does not hold in the traditional fixed design setting for
regression-type models. To deal with the estimation of these
covariate-dependent models, this paper presents a new score matching approach
for independent but not necessarily identically distributed data under a
general framework for both continuous and discrete responses, which includes a
novel generalized score matching method for count response regression. We prove
that our proposed score matching estimators are consistent and asymptotically
normal under mild regularity conditions. The theoretical results are supported
by simulation studies and a real-data example. Additionally, our simulation
results indicate that, compared to approximate maximum likelihood estimation,
the generalized score matching produces estimates with substantially smaller
biases in an application to doctoral publication data.
