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Please read our update regarding this competition, before applying, as some significant progress has been made (no winner yet as of 7/22). The purpose of the competition is to prove one of the mathematical conjectures recently discovered in our statistical research laboratory.
The conjecture is connected with the asymptotic distribution a new type of robust statistics, based on ranks, used in the definition of a new type of correlation and R-Square coefficients. This new type of coefficient is based on absolute values rather than squares, and leads to a much more robust (not sensitive to outliers) measurement when the number of observations, in a bucket of data, is above 100. It is of particular interest when processing big data. Proving the conjecture might require some good knowledge of permutation theory and combinatorics.
The first to prove or disprove our conjecture will win $500 and will have his name associated with the theorem in question - also known as the Third AnalyticBridge Theorem.
Our research laboratory occasionally comes up with new theorems that are particularly useful in the context of business analytics - in other words, theorems with direct, practical business applications. These theorems are derived from practical business problems rather than the other way around.
Here's what you need to prove:
Find an exact formula for the function q(n) defined in our recent article Correlation and R-Squared for Big Data, when c=1. Note that c=1 is the second easiest case after the standard case c=2; c=2 leads to the traditional correlation coefficient, where q(n)=n*(n^2-1)/6.
If this is too difficult, prove or disprove (for the fame only, no cash award offered) one of the following weaker results:
Any insights you have about the asymptotic distribution of q(n) would be great to share. Just computing q(n) alone for a specific n (larger than 15) is very tricky, as it requires either producing n! (factorial n) permutations, or do some very smart permutation sampling. Many hints are provided in my article to guide you. But so far (as of 7/10/2013), and to the best of my knowledge, there is no proof yet. Proving that there is no exact general (finite) formula is also accepted as a solution to this problem, and will also result in the $500 award.