= Rare-thickness leverage on inverse moments
\b[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 $\varepsilon$ with probability $p$, and the common thickness be $d_c$ otherwise. Define $a=1/\varepsilon$ and $b=1/d_c$. Among $n$ independent measurements the rare count $N$ has a <binomial distribution>, and the estimated mean reciprocal thickness is
$$
\widehat m=b+(a-b)\frac Nn.
$$
Consequently,
$$
\operatorname E(\widehat m)=b+(a-b)p,\qquad
\operatorname{Var}(\widehat m)=(a-b)^2\frac{p(1-p)}n.
$$
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 $p=0.0001$, $\varepsilon=0.001$ micrometres and $d_c=10.001$ micrometres. The true mean reciprocal thickness is approximately $0.19998\,\mathrm{\mu m^{-1}}$. With $n=1000$, zero rare observations has probability about 0.905 and gives $\widehat m\approx0.09999$. Exactly one rare observation instead gives $\widehat m\approx1.09989$. 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 requires
$$
n\ge\frac{\log(0.05)}{\log(1-p)},
$$
or 29,956 observations for this $p$. Estimating the rare fraction to relative <standard uncertainty> about 10% instead needs roughly one million independent observations, since $u(\widehat p)/p=\sqrt{(1-p)/(np)}$. 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.<NIST and SEMATECH e-Handbook of Statistical Methods>[§§1.3.6.6.11 and 1.3.6.6.9, density functions]{ref} Dimensional resolution and independent geometric bounds can be more informative than extending a simulation that never challenges its assumed tail.
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