Suppose you sample uniformly from the unit vectors in R^n. What are the distributions of the order statistics of the magnitudes of the components of the sampled vectors? That is, for 1 <= i <= n and x in [0,1], what is the probability that the i'th largest component of the vector (in absolute value) is less than or equal to x?
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$\begingroup$ It doesn't answer your question, but would it help to model the uniform distribution on the sphere by a random vector $(X_1/R, X_2/R, ..., X_n/R)$ where $X_1, \dots, X_N$ are i.i.d. standard Gaussians and R is defined to be $(X_1^2+\dots+X_n^2)^{1/2}$ ? Or have you tried this already? $\endgroup$– Yemon ChoiCommented Nov 20, 2009 at 8:19
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3 Answers
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There has been some work in the physics community on extreme statistics (i.e. distribution of largest and smallest components) of random vectors. See, link text for example. The largest component is approximately distributed like a Gumbel random variable, while the smallest component is approximately distributed like an exponential random variable.
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The distribution should be obtainable by integrating over the section of the simplex segment of the surface of the hypersphere bounded by the points (1,0,0,0,...), (1,1,0,0,0...)/sqrt(2), (1,1,1,0,0...)/sqrt(3) etc. along the ith axis.
All the distributions (n,m) have support contained within the unit interval, are piecewise smooth and share the same set of non-smooth points at the reciprocals of the square roots of the natural numbers.
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Your problem is closely related to order statistics for normal random variables, so you may find this paper useful: Percentage Points and Modes of Order Statistics from the Normal Distribution by Shanti S. Gupta
answered Nov 20, 2009 at 14:46