 Tail-model dependence of inferred density uncertainty

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.^[ref] At mean thickness $d$ and coefficient of variation $c$, the gamma model has shape $a=1/c^2$ and givesThe 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,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 $(0.9999{)}^{1000}\approx 0.905$. 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.