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Say we have samples $x_i$ from an unknown distribution $F(x)$. We want to know the number of samples $n$ such that we can say the distribution is "well sampled". Generally we need some sort of metric that is proportionally to "well-sampled"ness, and then we also need to be able to set some threshold that is "sufficient".

One idea I had for a metric is to take the average nearest-neighbor-distance ($d \equiv \langle \, \min(x_i - x_j)_j \, \rangle _i$ for $i ≠ j$) and divide by the standard deviation, $d/\sigma$. This ratio is always less than unity, but how to determine a threshold value is unclear.

If we were to construct an inferred estimate of the distribution, $f(x)$, e.g. using a KDE, then we could calculate a goodness-of-fit statistic (e.g. ks-test) between the CDF of $f(x)$ and that of the $x_i$ themselves. This approach has a more obvious way of deciding on a threshold value, but it requires numerous decisions to be made about how to construct the estimated $f(x)$ (e.g., for a KDE, a kernel and a bandwidth).

Are there any standard approaches or heuristics in the literature for this type of problem?

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