Manuscript En10-A@1 · En10-A.tex · remark · 0a1d7743

Remark: Confidence is not exact recovery

The finite Enter theorem and estimation contract · lines 944–970

A confidence statement is a repeated-sampling coverage guarantee. It is not a posterior probability without additional Bayesian assumptions. The Fisher-information comparison is not by itself a finite-shot confidence theorem.

The unclipped estimator

θ = \frac{Z - a}{b}

is unbiased and has variance

Var_{θ} (θ) = \frac{q _{θ} ( 1 - q _{θ} )}{n b ^{2}} = \frac{1}{n F _{A} ( θ )} .

Clipping may introduce bias, but decreases squared error relative to every $θ \in [0, 1]$ . Thus

E_{θ} (θ - θ)^{2} \leq \frac{q _{θ} ( 1 - q _{θ} )}{n b ^{2}} .

No unbiased-estimator assertion is made for the clipped estimator.

Displayed formulas in this passage

unlabeled · line 951
$θ = \frac{Z - a}{b}$
unlabeled · line 955
$Var_{θ} (θ) = \frac{q _{θ} ( 1 - q _{θ} )}{n b ^{2}} = \frac{1}{n F _{A} ( θ )} .$
unlabeled · line 964
$E_{θ} (θ - θ)^{2} \leq \frac{q _{θ} ( 1 - q _{θ} )}{n b ^{2}} .$

Remark: Confidence is not exact recovery

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