= Samplewise variance upper bound
{title2=$\widehat V\ge V_\theta\text{ almost surely}$}
A <samplewise variance upper bound> is a nonnegative statistic $\widehat V$ satisfying $\Pr_\theta\{\widehat V\ge V_\theta\}=1$, where $V_\theta=\operatorname{Var}_\theta(\widehat\theta)$ is the fixed sampling <variance> at parameter $\theta$. It controls each realized value outside a probability-zero exceptional set. This implies the expected-value condition for a <conservative variance estimator> when the expectation exists; the converse fails, as the parent article's counterexample shows. The primary sampling-design definition only asserts the weaker expectation condition.<Conservative variance estimation for sampling designs with zero pairwise inclusion probabilities by Peter M. Aronow and Cyrus Samii>[§3.1, Proposition 2]{ref}
Suppose $\widehat\theta$ is an <unbiased estimator> with finite <variance>, and let $0<\alpha<1$. Then <Chebyshev's inequality> and the samplewise comparison give
$$
\Pr_\theta\!\left\{|\widehat\theta-\theta|>\sqrt{\widehat V/\alpha}\right\}
\le\Pr_\theta\!\left\{|\widehat\theta-\theta|>\sqrt{V_\theta/\alpha}\right\}
\le\alpha.
$$
No independence between $\widehat V$ and the centre is needed. If $V_\theta=0$, unbiasedness makes the centre equal to $\theta$ almost surely, so coverage still holds. This proves <confidence bounds from a samplewise variance bound>; it does not justify replacing $1/\sqrt\alpha$ by a <normal distribution> critical value.
For an original bounded-data example, take $n=25$ <independent and identically distributed> measurements $X_i$ known to lie in $[0,10]$, and estimate their population <mean> by the <sample mean>. With $\mu=EX_i$, the inequality $X_i(10-X_i)\ge0$ gives
$$
\operatorname{Var}(X_i)\le10\mu-\mu^2\le25,
\qquad \operatorname{Var}(\overline X)\le25/25=1.
$$
The deterministic choice $\widehat V=1$ is a <samplewise variance upper bound> for every such population. Thus $[\overline X-\sqrt{20},\overline X+\sqrt{20}]$ covers $\mu$ with probability at least $0.95$. The radius $\sqrt{20}\approx4.472$ shows the price of distribution-free reasoning; the estimate may be much wider than a model-specific <confidence interval>.
Back to article page
