On the comparison of the Spearman and Kendall metrics between linear orders

This paper bears on the comparison of two well-known metrics between linear orders called the Kendall and Spearman metrics or/and of their normalized versions respectively known as the Kendall tau and the Spearman rho. Using a combinatorial approach based on the partial order intersection of the two compared linear orders, we first prove a relation between these two metrics and a semi-metric, equivalent to the classical Daniels inequality (1948) and to a Guilbaud formula (1980). Then, this approach allows to express the difference tau-rho as a simple function of parameters of this same partial order, to compute the maximum value of this difference and to characterize the corresponding pairs of linear orders. Finally, it also leads to discover an ordinal monotonicity property of the Spearman metric.