Hájek卷积定理的二阶扩展及其统计应用

中文摘要 AI 辅助

对于一类正则估计量，Hájek在其著名的“卷积定理”中表明，正则估计量的渐近分布是有效估计量分布与某个残差分布的卷积。这一结果构成了正则估计量渐近效率概念的基础。在本文中，我们提供了该经典结果的二阶版本。通过引入一类具有有效Edgeworth展开的二阶正则估计量，我们在邻近备择假设下推导了它们的渐近分布，并证明该分布是二阶有效分布与某个二阶残差分布的卷积。这构成了Hájek卷积定理的二阶扩展。基于此，我们为二阶正则估计量引入了“二阶稳健性”的概念。对于时间序列模型中一类一般的贝叶斯估计量和最小对比估计量，这种二阶稳健性被用于（i）刻画二阶稳健先验，（ii）比较极大似然估计量与Whittle估计量的二阶稳健性。

英文摘要

For a class of regular estimators, Hájek, in his celebrated ``Convolution Theorem,'' showed that the asymptotic distribution of a regular estimator is the convolution of the distribution of an efficient estimator and some residual distribution. This result constitutes the foundation of the concept of asymptotic efficiency of regular estimators. In this paper, we provide a second-order version of that classical result. Introducing a class of second-order regular estimators with a valid Edgeworth expansion, we derive their asymptotic distribution under contiguous alternatives and show that it is the convolution of the second-order efficient distribution and some second-order residual distribution. This constitutes a second-order extension of Hájek's convolution theorem. Based on this, we introduce a concept of {\it second-order robustness} for second-order regular estimators. For a class of general Bayes estimators and minimum contrast estimators in time series models, this second-order robustness is used (i) in the characterization of second-order robust priors, (ii) in a comparison between the second-order robustness of maximum likelihood and Whittle estimators.

发表机构

• Nanzan University（南山大学）
• Waseda University（早稻田大学）
• Institute of Information Theory and Automation, Czech Academy of Sciences（捷克科学院信息理论与自动化研究所）

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