 Confidence bounds from a samplewise variance bound

Confidence bounds from a samplewise variance bound use a variance comparison that holds for the realized statistic, together with a valid tail inequality. The unbiased two-sided construction is proved in samplewise variance upper bound. A useful extension includes a separately justified bias budget: assumewhere $B\ge 0$ is a validated deterministic bound and the estimator has finite variance. For $0<\alpha <1$, defineIf $\theta$ lies outside this interval, the triangle inequality implies $|\hat{\theta }-E_{\theta }\hat{\theta }|>\sqrt{\hat{V}/\alpha }$. The samplewise bound and Chebyshev's inequality then give ${\mathrm{Pr}}_{\theta }(\theta \notin I)\le \alpha$. No independence between the centre and $\hat{V}$ is required. Zero true variance is covered because the centre then equals its expectation almost surely.

For an original measurement example in a fixed unit, suppose the reported estimate is $2.00$, a validated residual-bias bound is $B=0.20$, and a deterministic samplewise variance upper bound is $0.04$. With $\alpha =0.05$, the radius isso the interval is approximately $[0.9056,3.0944]$. The conclusion applies over repeated measurements under those assumptions. The particular realized interval is not assigned a frequentist probability that the fixed parameter is inside it. Replacing the radius by $1.96\sqrt{0.04}=0.392$ would discard both the bias allowance and the tail justification.

The bound $B$ is an assumption about the centre's possible systematic displacement; a guessed correction uncertainty is not automatically such a bound. NIST distinguishes the coverage interpretation from merely applying a conventional multiplier to a standard uncertainty.^[ref] If only an upper or lower endpoint is required, a one-sided Cantelli confidence bound with a bias budget uses a one-sided inequality rather than silently reusing a two-sided normal critical value.