🤖 AI Summary
Classical point estimation theory suffers from fundamental limitations, including parameter dependence, undefined estimators at boundary points, the absence of unbiased estimators in certain settings, and a lack of rigorous theoretical grounding for maximum likelihood estimation. This work reframes estimators as functions on the parameter space and, adopting Fisher’s perspective of continuous null hypotheses within the Bahadur framework, introduces a generalized estimation mapping that coherently resolves these issues. By leveraging Hilbert space geometry and score function analysis, the approach elevates maximum likelihood to an exact result concerning the score function, naturally yielding the Cramér–Rao bound and sufficiency. It also provides valid estimates at boundary samples, demonstrates the nonexistence of uniformly minimum-variance unbiased estimators, and derives the Fisher information bound in a unified, streamlined manner—subsuming classical results as special cases.
📝 Abstract
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.