Transform martingale estimating functions

TL;DR

This paper introduces a novel estimation approach for discrete time stochastic processes using transform-based martingale estimating functions, enabling inference even when likelihoods are intractable.

Contribution

It develops a new class of transform-based martingale estimating functions within the quasi-likelihood framework, extending applicability to processes with infinite second moments.

Findings

Provides a method to approximate score functions using integral transforms.

Extends quasi-likelihood methods to a broader class of stochastic processes.

Enables estimation without explicit likelihood functions.

Abstract

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.