AI 中文总结
该论文重新审视经典估计问题，以巴哈杜尔的理论为基础，在其理论难点处做小修改，如让估计量成为参数空间上的函数。通过这些修改逐讲追踪收益，包括解决估计量存在性问题、解释一些准则出现的原因等，未推翻经典，而是用费希尔的特征解释经典工具。
AI 中文摘要
巴哈杜尔的《估计理论讲座》在希尔伯特空间几何中发展了经典点估计理论，并诚实地记录了该理论的难点：局部最优无偏估计依赖于参数，两点参数空间产生的估计被巴哈杜尔称为荒谬的，二项式抽样中的优势比没有无偏估计，最大似然的优点只是启发式的。我们以巴哈杜尔的符号和发展呈现讲座的一个子集，并在每个难点处做一个小修改：对于样本空间中的每个值，估计量τ成为参数空间上的函数而非其中的一个点。逐讲追踪收益：在点估计不存在的边界样本处存在的估计量；一个基本引理表明没有逐点准则允许一致最优估计量，解释了为什么可容许性、极小极大性、贝叶斯平均和无偏性会出现；通过信息Λ(τ)进行评估，得分通过一个三行论证均匀达到费希尔信息界；在从点估计量到广义估计量的两个映射下，克拉默 - 拉奥达到性和充分性作为该界的等式情况被恢复；最大似然启发式转化为关于得分的精确陈述。没有推翻任何经典内容；使用费希尔将估计描述为连续显著性检验的特征来解释经典工具。
英文摘要
Bahadur's \emph{Lectures on the Theory of Estimation} develop the classical theory of point estimation inside the geometry of Hilbert space, and they record with unusual honesty where the theory strains: the locally best unbiased estimate depends on the parameter, a two-point parameter space yields an estimate Bahadur calls absurd, the odds ratio in binomial sampling has no unbiased estimate, and the virtues of maximum likelihood enter as heuristics and remain heuristics. We present a subset of the lectures, in Bahadur's notation and development, and at each strain make one small modification: for each value in the sample space, an estimate $τ$ becomes a function on the parameter space rather than a point in it, the continuum of null hypotheses that Fisher described in 1955. Bahadur's own definition of an estimate, square-integrable at every distribution in the family, already supplies the domain. The payoffs are tracked lecture by lecture: estimators that exist at boundary samples where point estimates do not; an elementary lemma showing that no pointwise criterion admits a uniformly optimal estimator, which explains why admissibility, minimaxity, Bayes averaging, and unbiasedness arose as responses; assessment by information, $Λ(τ)$, with the score attaining the Fisher information bound uniformly by a three-line argument; Cramér--Rao attainment and sufficiency recovered as equality cases of that bound under two maps from point estimators to generalized estimators; and the maximum likelihood heuristics converted into exact statements about the score. Nothing classical is overturned; the classical apparatus is explained using Fisher's characterization of estimation as a continuum of significance tests.
Comments21 pages, no figures
