Parametric estimation. Finite sample theory
参数估计的有限样本理论AI 解读 ↓
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参数估计的有限样本理论
- 研究问题
- 在样本量固定且参数模型可能误设时，如何建立似然估计与统计推断的定量理论，并处理较大或增长的参数维度？
- 方法与思路
- 作者以非渐近方式重新考察局部渐近正态理论，建立准最大似然的大偏差界及对数似然过程的局部二次夹逼。
- 材料陈述的贡献
- 摘要称，该框架给出集中、置信与风险界及估计量展开，并可导出经典渐近结果和达到根号样本量精度的样本量下界。
- 阅读关注点
- 应核对误设下的目标参数、局部二次夹逼条件，以及维度、样本量与误差界的关系，确认效率结论适用的设定。
- 阅读边界
- 仅据版本未注明的OpenAlex摘要；具体条件与样本量下界未提供，适用于任意参数维度不意味着无条件获得根号样本量精度。
原文摘要
The paper aims at reconsidering the famous Le Cam LAN theory. The main features of the approach which make it different from the classical one are as follows: (1) the study is nonasymptotic, that is, the sample size is fixed and does not tend to infinity; (2) the parametric assumption is possibly misspecified and the underlying data distribution can lie beyond the given parametric family. These two features enable to bridge the gap between parametric and nonparametric theory and to build a unified framework for statistical estimation. The main results include large deviation bounds for the (quasi) maximum likelihood and the local quadratic bracketing of the log-likelihood process. The latter yields a number of important corollaries for statistical inference: concentration, confidence and risk bounds, expansion of the maximum likelihood estimate, etc. All these corollaries are stated in a nonclassical way admitting a model misspecification and finite samples. However, the classical asymptotic results including the efficiency bounds can be easily derived as corollaries of the obtained nonasymptotic statements. At the same time, the new bracketing device works well in the situations with large or growing parameter dimension in which the classical parametric theory fails. The general results are illustrated for the i.i.d. setup as well as for generalized linear and median estimation. The results apply for any dimension of the parameter space and provide a quantitative lower bound on the sample size yielding the root-n accuracy.