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 wr

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!