= One-sided Cantelli confidence bound with a bias budget
A <one-sided Cantelli confidence bound with a bias budget> combines a validated bound on the estimator's systematic displacement with a one-sided <variance> inequality. Suppose $|E_\theta\widehat\theta-\theta|\le B$, where $B\ge0$ is deterministic, and a nonnegative statistic satisfies $\widehat V\ge V_\theta=\operatorname{Var}_\theta(\widehat\theta)$ almost surely. For $0<\alpha<1$, define
$$
L=\widehat\theta-B-\sqrt{\frac{1-\alpha}{\alpha}\widehat V}.
$$
Then $\Pr_\theta(\theta\ge L)\ge1-\alpha$. This is a lower <confidence bound>, with an unbounded upper side.
For the proof, put $Y=\widehat\theta-E_\theta\widehat\theta$. If $\theta<L$, the <bias> bound implies
$$
Y>\sqrt{\frac{1-\alpha}{\alpha}\widehat V}
\ge\sqrt{\frac{1-\alpha}{\alpha}V_\theta}.
$$
For positive $V_\theta$, <Cantelli's inequality> bounds this event by $V_\theta/(V_\theta+a^2)=\alpha$ at the last displayed threshold $a$. For zero <variance>, $Y=0$ almost surely and the conclusion follows directly. A random $\widehat V$ may depend on the centre: the samplewise comparison is what supports the event inclusion.
For an original numerical bound, let $\widehat\theta=2.0$, $B=0.2$, $\widehat V=0.04$, and $\alpha=0.1$. The square-root term is $0.6$, so $L=1.2$ is a 90 percent lower bound under these assumptions. Reversing the inequality gives the separately valid upper bound $U=2.8$. Each has 90 percent one-sided coverage; reporting both endpoints together only guarantees 80 percent by the <union bound>. To guarantee a joint 90 percent interval this way, allocate error probabilities summing to $0.1$ across its two sides.
This construction complements the two-sided result in <confidence bounds from a samplewise variance bound>. It needs a <bias> bound and an almost-sure <variance> bound, not merely a <conservative variance estimator> in the expected-value sense. A coverage interpretation must follow the actual tail argument, as emphasized by NIST's discussion of expanded 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}
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