This chapter gives a survey of the basic concepts of estimating functions, which are used in subsequent chapters. The concept of unbiasedness for estimating functions is introduced as a generalization of the concept of an unbiased estimator. Godambe efficiency, also known as the Godambe optimality criterion, is introduced by generalizing the concept of minimum variance unbiased estimation. Within the class of estimating functions which are unbiased and information unbiased, the score function is characterized as the estimating function with maximal Godambe efficiency. Extensions to the multiparameter case are given, and the connection to the Riesz representation theorem is described briefly. This chapter also discusses a number of examples from semiparametric models, martingale estimating functions for stochastic processes, empirical characteristic function methods and quadrat sampling; the estimating equations in some of these examples have possibly more than one solution.
Keywords: empirical characteristic function, Godambe efficiency, Godambe optimality criterion, information unbiasedness, martingale estimating function, minimum variance unbiased estimation, quadrat sampling, Riesz representation theorem, score function, unbiasedness
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