= Coverage failure under random variance-ratio limits
<Coverage failure under random variance-ratio limits> can persist even when the centre is exactly normal, the <variance> estimate is conservative in expectation, and its absolute error tends to zero. What fails is relative-scale concentration. Let $Z\sim N(0,1)$ and a positive random scale $R$ be independent, and define
$$
\widehat\theta_n=\theta+\frac Z{\sqrt n},\qquad
V_n=\frac1n,\qquad\widehat V_n=\frac Rn.
$$
The <variance> ratio is $R$ at every sample size. Its nondegenerate distribution need not approach the constant required in <asymptotic conservative Wald coverage under variance-ratio convergence>.
Conditional on $R$, a normal critical value $z>0$ gives interval coverage $f(R)=2\Phi(z\sqrt R)-1$. Therefore unconditional coverage is $Ef(R)$ under the independence assumption. A useful shape calculation is
$$
f''(r)=-\frac{z\phi(z\sqrt r)}{2r^{3/2}}(1+z^2r)<0,\qquad r>0.
$$
The function is strictly concave. If $ER=1$, Jensen's inequality puts $Ef(R)\le f(1)$, with strict inequality for a nonconstant scale. An expectation equality therefore does not reproduce the normal interval's nominal coverage.
For an original strictly conservative example, let $R$ take $0.25$ and $2.75$ with equal probabilities. Then $E\widehat V_n=1.5V_n$, but at nominal 95 percent the exact coverage is
$$
\frac12\{2\Phi(1.959964\sqrt{0.25})-1\}
+\frac12\{2\Phi(1.959964\sqrt{2.75})-1\}
\approx0.83588.
$$
This value is unchanged as $n$ grows. Meanwhile $\widehat V_n-V_n=(R-1)/n$ tends to zero, showing why absolute consistency at a vanishing <variance> scale is insufficient. The normal cdf evaluation uses the pinned primary numerical definition.<SciPy 1.16.2 documentation>[scipy.stats.norm, cdf and ppf]{ref}
If the scale depends on the standardized error, the conditional normal formula above cannot simply be retained; the joint law is needed. NIST likewise distinguishes a justified coverage probability from selecting a conventional uncertainty multiplier.<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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