# How to measure “well-sampled”ness i.e. how well a distribution is sampled in a given set of data

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?