Smoothing complements and randomized score functions
Let \(({\mathcal X},{\mathcal A})\) be a measurable space and \(\{P_ \theta, \theta\in\Theta\}\) be a family of probability measures defined on \(({\mathcal X},{\mathcal A})\). Let \(T(\cdot)\) be a real-valued measurable function on \({\mathcal X}\) and define \(g(\theta)=E_ \theta[\Gamma(x)]\). Assume that \(g(\theta)\) is differentiable with respect to \(\theta\) and denote the derivative by \(g'(\theta)\). This paper studies connections between the infinitesimal perturbation analysis (IPA) and the likelihood ratio or score function method for estimation of \(g(\theta)\). The author shows that likelihood ratio derivative estimators are IPA estimators obtained through a special construction based on the notion of multiplicative smoothing. (Reviewer's remarks: The concepts used in this area seem to differ from the standard definitions in the areas of statistics and probability. This is illustrated for instance by the fact that estimators cannot be considered as a function of the unknown parameter \(\theta\) as in classical statistics, see page 42, line 3.) Multiplicative smoothing in some sense enlarges the probability space whereas conditioning ``shrinks'' it.
scientific article; zbMATH DE number 165329