 Rare-thickness leverage on inverse moments

Rare-thickness leverage on inverse moments is the disproportionate contribution of uncommon, very thin layers to inferred density when $\rho =K/\delta$. Its sampling consequence is especially revealing: a sample can miss the influential component and report very little apparent variability.

Consider an original two-point thickness model. Let the rare thickness be $\epsilon$ with probability $p$, and the common thickness be $d_c$ otherwise. Define $a=1/\epsilon$ and $b=1/d_c$. Among $n$ independent measurements the rare count $N$ has a binomial distribution, and the estimated mean reciprocal thickness isConsequently,Multiplication by $K$ converts these into the mean and variance of the sample mean inferred density. The formulas are exact for this constructed model.

Take $p=0.0001$, $\epsilon =0.001$ micrometres and $d_c=10.001$ micrometres. The true mean reciprocal thickness is approximately $0.19998\mu {\mathrm{m}}^{-1}$. With $n=1000$, zero rare observations has probability about 0.905 and gives $\hat{m}\approx 0.09999$. Exactly one rare observation instead gives $\hat{m}\approx 1.09989$. The sampling distribution is highly asymmetric despite the large-looking sample count. When every sampled value is the same common thickness, the ordinary sample variance is even zero; it has not revealed the missing component.

Rare-event discovery versus inverse-moment precision distinguishes two experimental objectives. To have at least 95% probability of seeing one rare event requiresor 29,956 observations for this $p$. Estimating the rare fraction to relative standard uncertainty about 10% instead needs roughly one million independent observations, since $u(\hat{p})/p=\sqrt{(1-p)/(np)}$. Seeing one event is not precise quantification of its contribution.

A distributional assumption therefore needs physical support near zero thickness, not just a fit to the sample centre. NIST's gamma distribution and lognormal distribution entries specify distinct lower-tail models; choosing either is a modeling decision.^[ref] Dimensional resolution and independent geometric bounds can be more informative than extending a simulation that never challenges its assumed tail.