Semiparametric Statistics and the Spread Inequality

Classical mathematical statistics deals with models that are parametrized by a Euclidean, i.e. finite dimensional, parameter. Quite often such models have been and still are chosen in practical situations for their mathematical simplicity and tractability. However, these models are typically inappropriate since the implied distributional assumptions cannot be supported by hard evidence. It is natural then to relax these assumptions. This leads to the class of semiparametric models. These models have been studied in a local asymptotic setting, in which the Convolution Theorem yields bounds on the performance of regular estimators. Alternatively, local asymptotics can be based on the Local Asymptotic Minimax Theorem and on the Local Asymptotic Spread Theorem, both valid for any sequence of estimators. This Local Asymptotic Spread Theorem is a straightforward consequence of a Finite Sample Spread Inequality, which has some intrinsic value for estimation theory in general. We will discuss both the Finite Sample and Local Asymptotic Spread Theorem, as well as the Convolution Theorem.