Spoken like a true statistician! Yes, the
statisticians keep assuming Gaussian, fitting
distributions to data, etc., and you have
a way out: Of course it's wrong, but it's
still useful! Wow!

> If you have a paper to suggest that general high-dimensional order statistics makes sense, throw it this way.

Do an hypothesis test. For positive integers
m and n, consider m > 1 points in real Euclidean n-space.
We want to know if point m is distributed like
the rest. Our null hypothesis is that all the
points have the same distribution and are i.i.d.

Now for positive integer k, for each of the m points,
we can find the distance to the farthest of the
k nearest neighbors. So, we get m distances.
These, however, are not i.i.d. But can do a little
work to show have a finite group of measure
preserving transformations such that, if we sum over the
group, then we get something similar to a permutation
test. So, if the distance for point m is
in the upper 2% of the distances, then
have probability of type I error of 2% and, thus,
an hypothesis test. So, have a multidimensional,
distribution-free hypothesis test. That is, intuitively if point m is 'too far away' from the other m - 1 points, then we reject the null hypothesis that point m is distributed like the other m - 1 points (we continue to believe in the independence assumption and do not reject it). But the
k-nearest neighbors with the Euclidean norm
need not be the only one used -- so get
a family of such tests.

Intuitively, for n = 2, we have a density
that looks like, say, a mountain range.
Suppose the rocks of the range are porous
to water and pour in some water. Then
the water all seeks the same level. So,
we have multiple lakes, with islands, with
lakes, etc. Suppose the water covers 2%
of the probability mass. Then the test is
to see if point m falls in the water.

Yes, we expect the lake boundaries to be
fractals. So, right, with enough points m,
we are approximating the fractal boundaries
of the lakes. Increasing k makes the
approximation to the boundaries more smooth.
If pour in more water, then we get another
set of contours. So, we get a way to do
contour maps of the density. So, we get
a technique of multidimensional density
estimation.

E.g., go to a beach, pick up m = 200 rocks,
and for each measure n = 10 properties. Now
given one more rock, ask if it came from that
beach!

As I recall, there is such a paper in
'Information Sciences' in 1999 about
an "anomaly detector'. Why 'Information
Sciences'? Because the paper suggested
monitoring large server farms and networks
for 'health and wellness' by this technique.
So, we would get multidimensional, distribution-free
'behavioral monitoring' with false alarm rate
we could set in advance and get exactly.
The work does not promise to have the best
power in the sense of the Neyman-Pearson
result, but the paper uses Ulam's 'tightness'
to argue that the test is not 'trivial'.
I have yet to see a distribution-free
test that tries to argue it is as powerful
as Neyman-Pearson!

While we don't get Neyman-Pearson, we
get an approximation to the smallest
region where we will make a type II error
and, thus, in a sense for any alternative
distribution for point m the most power
in the goofy sense of shifting that
alternative distribution around! The
argument is just a simple use of Fubini's theorem! So,
in a goofy but possibly "useful" sense
we minimize type II error. There might
be a duality situation here!