A note on the relationship between Spearman's ρρ and Kendall's ττ for extreme order statistics

This note focuses on the relationship between Spearman's ρnρn and Kendall's τnτn for the two extreme order statistics X(1)X(1) and X(n)X(n) of n   independent and identically distributed continuous random variables. We present three new formulas for computing Spearman's ρnρn. One of the formulas leads to a recursion relation. We use this recursion relation to establish inequality relationships between ρnρn and τnτn. The recursion relation also provides an alternative proof of the result that the sequence of ratios ρn/τnρn/τn converges to 32 as the sample size n goes to infinity.