  A composite likelihood is an inference function derived by multiplying a set
of likelihood components. This approach provides a flexible framework for
drawing inference when the likelihood function of a statistical model is
computationally intractable. While composite likelihood has computational
advantages, it can still be demanding when dealing with numerous likelihood
components and a large sample size. This paper tackles this challenge by
employing an approximation of the conventional composite likelihood estimator,
which is derived from an optimization procedure relying on stochastic
gradients. This novel estimator is shown to be asymptotically normally
distributed around the true parameter. In particular, based on the relative
divergent rate of the sample size and the number of iterations of the
optimization, the variance of the limiting distribution is shown to compound
for two sources of uncertainty: the sampling variability of the data and the
optimization noise, with the latter depending on the sampling distribution used
to construct the stochastic gradients. The advantages of the proposed framework
are illustrated through simulation studies on two working examples: an Ising
model for binary data and a gamma frailty model for count data. Finally, a
real-data application is presented, showing its effectiveness in a large-scale
mental health survey.
