Assuming I have a data set of known size, and there is one object that I want to test for being in an approximately uniform distributed region of the data set.
For the query object, I know the $k$ nearest neighbors and the associated distances. If the data set is uniform there, the distances should be $c\cdot\chi(d)$ distributed, I believe (with $c$ a constant, representing some kind of data set diameter).
So I have a spherical subset of the total data set that contains $k$ objects and has a known radius of $d_k$, the distance to the $k$th object, or equally the maximum distance within the $k$ nearest neighbors.
Note that the $d_k$s - if I would be looking at more than one such subset - likely are not $\chi(d)$ distributed, as they are derived from the distances. Is there any assumption that I can make on the distribution of the $d_k$ for a dataset?
So what is the best way to test if the object is in a somewhat uniformly distributed area of the data set (should I try a Kolmogorov-Smirnov goodness of fit test to $\chi(d)$ of the known distances? But I'm probably not seeing the complete distribution!), and what is the best guess at the density? My key idea is that the object might be an outlier if is in a low-density and approximately uniform area of the data set. Methods such as LOF fail here, because the outlier is too far out. LOF works good for detecting outliers nearby the actual data, but not within a uniform sparse area.