= Tail-model dependence of inferred density uncertainty
\b[Tail-model dependence of inferred density uncertainty] is the sensitivity of a <density> estimate's distribution to the assumed probability of very small positive thicknesses. For $\rho=K/\delta$ with fixed $K=M/A>0$, a tiny thickness has disproportionate influence through $1/\delta$. The main body of a thickness histogram can look stable while the inferred mean or <variance> remains model-dependent.
A <gamma distribution> and a <lognormal distribution> provide useful explicit models. NIST specifies their density functions and parameter conventions.<NIST and SEMATECH e-Handbook of Statistical Methods>[§§1.3.6.6.11 and 1.3.6.6.9]{ref} At mean thickness $d$ and <coefficient of variation> $c$, the gamma model has shape $a=1/c^2$ and gives
$$
\operatorname E(1/\delta)=\frac1{d(1-c^2)},\qquad c<1.
$$
The lognormal model gives $(1+c^2)/d$. The two agree closely for small $c$, but that agreement concerns those model families, not every positive distribution with a similar spread.
An original rare-event example makes the limitation explicit. Let thickness be $0.001$ micrometres with probability $0.0001$ and $10.001$ micrometres otherwise. Mean thickness is exactly 10 micrometres and its <standard deviation> is approximately 0.10 micrometres, about 1% of the mean. Nevertheless,
$$
\operatorname E(1/\delta)=\frac{0.0001}{0.001}+\frac{0.9999}{10.001}\approx0.19998\,\mathrm{\mu m^{-1}},
$$
nearly twice the reciprocal of mean thickness. The rare component supplies about half the inverse moment. These are deliberately constructed model values, not measured polymer-layer dimensions.
<Rare-thickness leverage on inverse moments> can escape an apparently reassuring sample. With 1,000 independent thickness observations, the chance of seeing none of that rare component is $(0.9999)^{1000}\approx0.905$. A finite simulation or a smooth fit to common observations need not validate the unobserved lower tail.
A useful experiment investigates whether dimensional resolution, specimen geometry or physical constraints rule out such near-zero layers. Report sensitivity to defensible alternative lower-tail models or measured lower bounds. Choosing a positive distribution guarantees positive thickness, but does not establish that its inverse moments describe the measurement process.
\Image[/rare-thickness-density-leverage.png]
{title=An original two-point thickness model: a component appearing in only one of ten thousand observations supplies about half the expected reciprocal thickness, and therefore half the expected inferred density at fixed mass per area.}
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