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.

翻译：经典数理统计处理的是由欧几里得参数（即有限维参数）参数化的模型。这类模型因其数学简洁性和易处理性，在过去和现在的实际应用中常常被选用。然而，这些模型通常并不恰当，因为隐含的分布假设缺乏确凿证据支持。于是，自然地需要放松这些假设。这便引出了半参数模型类。这类模型已在局部渐近框架下得到研究，其中卷积定理为正则估计量的性能提供了界。另一种方式，局部渐近分析可基于局部渐近极小极大定理和局部渐近离差定理，两者均对任意估计量序列成立。局部渐近离差定理是有限样本离差不等式的直接推论，该不等式本身对一般估计理论具有内在价值。本文将讨论有限样本离差定理与局部渐近离差定理，以及卷积定理。