  An estimation method is proposed for a wide variety of discrete time
stochastic processes that have an intractable likelihood function but are
otherwise conveniently specified by an integral transform such as the
characteristic function, the Laplace transform or the probability generating
function. This method involves the construction of classes of transform-based
martingale estimating functions that fit into the general framework of
quasi-likelihood. In the parametric setting of a discrete time stochastic
process, we obtain transform quasi-score functions by projecting the
unavailable score function onto the special linear spaces formed by these
classes. The specification of the process by any of the main integral
transforms makes possible an arbitrarily close approximation of the score
function in an infinite-dimensional Hilbert space by optimally combining
transform martingale quasi-score functions. It also allows an extension of the
domain of application of quasi-likelihood methodology to processes with
infinite conditional second moment.
