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Transform martingale estimating functions. (English) Zbl 1126.62074

Summary: 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.

MSC:

62M09 Non-Markovian processes: estimation
60E10 Characteristic functions; other transforms
62G20 Asymptotic properties of nonparametric inference
62M99 Inference from stochastic processes
60G42 Martingales with discrete parameter
62M05 Markov processes: estimation; hidden Markov models
46N30 Applications of functional analysis in probability theory and statistics
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