order statistics for components of a random unit vector - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T02:18:53Z http://mathoverflow.net/feeds/question/6233 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6233/order-statistics-for-components-of-a-random-unit-vector order statistics for components of a random unit vector Jeff Hussmann 2009-11-20T07:31:20Z 2009-12-18T19:22:22Z <p>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 &lt;= i &lt;= 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? </p> http://mathoverflow.net/questions/6233/order-statistics-for-components-of-a-random-unit-vector/6244#6244 Answer by QuantumBrian for order statistics for components of a random unit vector QuantumBrian 2009-11-20T10:13:47Z 2009-11-20T10:13:47Z <p>There has been some work in the physics community on extreme statistics (i.e. distribution of largest and smallest components) of random vectors. See, <a href="http://arxiv.org/abs/0708.0176/" rel="nofollow" title="ArXiv:0708.0176">link text</a> 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.</p> http://mathoverflow.net/questions/6233/order-statistics-for-components-of-a-random-unit-vector/6272#6272 Answer by John D. Cook for order statistics for components of a random unit vector John D. Cook 2009-11-20T14:46:48Z 2009-11-20T14:46:48Z <p>Your problem is closely related to order statistics for normal random variables, so you may find this paper useful: <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.aoms/1177704982&amp;page=record" rel="nofollow">Percentage Points and Modes of Order Statistics from the Normal Distribution</a> by Shanti S. Gupta </p> http://mathoverflow.net/questions/6233/order-statistics-for-components-of-a-random-unit-vector/7801#7801 Answer by Jack Evans for order statistics for components of a random unit vector Jack Evans 2009-12-04T19:12:55Z 2009-12-04T19:12:55Z <p>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.</p> <p>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. </p>