Explicit Expressions for the first 5 moments of Spearman's RHO and a partial(elementary!) proof of its Asymptotic Normality
By Shalosh B. Ekhad
Let's define the following "measure of disarray" on permutations of length n, let's call it RHO(pi)
n
-----
\ 2
RHO(pi) = ) (pi[i] - i)
/
-----
i = 1
The expectation of RHO is
n (n - 1) (n + 1)
-----------------
6
and in Maple notation
1/6*n*(n-1)*(n+1)
The variance of RHO is
2 2
n (n - 1) (n + 1)
-------------------
36
and in Maple notation
1/36*n^2*(n-1)*(n+1)^2
The, 3, -th moment about the mean of RHO is
0
and in Maple notation
0
The, 4, -th moment about the mean of RHO is
3 3 2 3
n (n - 1) (25 n - 38 n - 35 n + 72) (n + 1)
-----------------------------------------------
10800
and in Maple notation
1/10800*n^3*(n-1)*(25*n^3-38*n^2-35*n+72)*(n+1)^3
The, 5, -th moment about the mean of RHO is
0
and in Maple notation
0
Let's now look at the scaled moment
The limit of the, 3, -th scaled moment about the mean of RHO is
0
The limit of the, 4, -th scaled moment about the mean of RHO is
3
more precisely, the asymptotic is
114 6 102 6 102 6 102 1
3 - ---- - ---- + ----- - ---- + ----- - ---- + ----- + O(----)
25 n 2 3 4 5 6 7 8
5 n 25 n 5 n 25 n 5 n 25 n n
and in Maple notation
3-114/25/n-6/5/n^2+102/25/n^3-6/5/n^4+102/25/n^5-6/5/n^6+102/25/n^7+O(1/n^8)
The limit of the, 5, -th scaled moment about the mean of RHO is
0
This proves asymptotic normality up to the 5th moment in a fully elementary \
way, confirming the fact that RHO is, most probably, asymptotically norm\
al