Contribution-weighted precision target for inverse moments ($n\ge {\theta }^2(1-p)/({\eta }^{2}p)$)

A contribution-weighted precision target for inverse moments chooses sample size according to how strongly rare observations affect the desired inverse moment, rather than according to the probability of seeing any rare observation. For positive thickness $H$, an inverse moment such as $\mu =\mathbb{E}(1/H)$ can be sensitive to very small thicknesses.

In an original two-point model, let reciprocal thickness be $a$ with rare probability $p$, and $b$ otherwise, with $a>b>0$. Then $\mu =b+(a-b)p$. For $n$ independent observations and rare count $N$, the estimate is $\hat{\mu }=b+(a-b)N/n$. Since $N$ has a binomial distribution,Define the fraction of the mean supplied by the rare excess over the baseline as $\theta =(a-b)p/\mu$. Its relative standard uncertainty isfor target relative uncertainty $\eta$. Here $0<\theta <1$. Rarity alone does not specify the required precision because the rare value's magnitude also enters.

For stipulated $p=10^{-4}$ and $\eta =0.10$, a rare-excess contribution $\theta =0.05$ requires at least 2,500 observations, whereas $\theta =0.50$ requires 249,975. A tenfold increase in contribution raises this variance-based count a hundredfold. The first case still has approximately $(1-10^{-4}{)}^{2500}=0.779$ probability of no rare observation. That is compatible with its standard-uncertainty target because missing the rare excess produces only a 5% relative error in that particular model.

This is not a 95% probability guarantee of a specified relative error. Discovery probability $1-(1-p{)}^n$, estimator variance and interval coverage are different criteria. Exact binomial distribution tail calculations address count probabilities; they do not turn a standard deviation into a coverage statement automatically.^[ref]

The calculation also treats $a$, $b$ and $p$ as specified planning quantities. Uncertainty in rare-event probability, imaging resolution or the reciprocal values requires further modeling; spatial dependence reduces the usefulness of an independent-sample count. In particular, increasing $n$ cannot establish an assumed physical lower-thickness bound that the measurement system never resolves.