= 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: assume
$$
|E_\theta\widehat\theta-\theta|\le B,\qquad
\widehat V\ge\operatorname{Var}_\theta(\widehat\theta)\quad\text{almost surely},
$$
where $B\ge0$ is a validated deterministic bound and the estimator has finite <variance>. For $0<\alpha<1$, define
$$
I=\left[\widehat\theta-B-\sqrt{\widehat V/\alpha},\ \widehat\theta+B+\sqrt{\widehat V/\alpha}\right].
$$
If $\theta$ lies outside this interval, the <triangle inequality> implies $|\widehat\theta-E_\theta\widehat\theta|>\sqrt{\widehat V/\alpha}$. The samplewise bound and <Chebyshev's inequality> then give $\Pr_\theta(\theta\notin I)\le\alpha$. No independence between the centre and $\widehat V$ 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 is
$$
0.20+\sqrt{0.04/0.05}\approx1.0944,
$$
so 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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[§§6.2–6.3]{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.
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