Leading Asympotitcs to order 3 of the moments of the Spearman's Footrule



                              By Shalosh B. Ekhad





Thereom: The first leading terms in the expression of the (2r)-th moment of \
    Spearman's Footrule are



                      /           2
      r         (3 r) |    r (20 r  - 501 r + 1363)             5            4
(2/45)  (2 r)! n      |1 + ------------------------ + r (57200 r  - 8425560 r
                      \             882 n

                  3              2                               /
     + 141955229 r  - 528900858 r  + 526923659 r + 2204496102)  /  (667458792
                                                               /

       \
     2 |   /   r
    n )|  /  (2  r!)
       / /



                             and in Maple notation



(2/45)^r*(2*r)!/(2^r)/r!*n^(3*r)*(1+1/882*r*(20*r^2-501*r+1363)/n+1/667458792*r
*(57200*r^5-8425560*r^4+141955229*r^3-528900858*r^2+526923659*r+2204496102)/n^2
)


   Also the first two terms in the exrpression for the (2r+1)-th moments are



      r         (3 r + 1)
(2/45)  (2 r)! n

    /                                    3          2                   \
    |  2 r (2 r + 1)   r (2 r + 1) (220 r  - 15873 r  + 39464 r + 63507)|   /
    |- ------------- - -------------------------------------------------|  /  (
    \       63                             916839 n                     / /

     r
    2  r!)



                             and in Maple notation



(2/45)^r*(2*r)!/(2^r)/r!*n^(3*r+1)*(-2/63*r*(2*r+1)-1/916839*r*(2*r+1)*(220*r^3
-15873*r^2+39464*r+63507)/n)


Note that this immediately implies asymptotic  normality, as first proved by\
     Persin Diaconis, in a joint paper with Ron Graham



 Spearman's Footrule as a measure of disarray,  J. Royal Stat.  Soc., Sectio\
    n B, vol. 39, No. 2 (1977), pp. 262-268 . See Theorem 1



                              but gives much more!