A Comparison of Correlation Coefficients via a Three-Step Bootstrap Approach

In this article we compare ten correlation coefficients using a three-step bootstrap approach (TSB). A three-step bootstrap is applied to determine the optimal repetitions, $B$, to estimate  the standard error of the statistic with certain degree of accuracy. The coefficients in question are Pearson product moment ($r$), Spearman's rho ($ho$), Kendall's tau ($au$) , Spearman's Footrule ($F_t$),  Symmetric Footrule ($C$), the Greatest deviation ($R_g$), the Top - Down ($r_T$), Weighted Kendall's tau ($au_w$), Blest ($u$), and Symmetric Blest's coefficient ($u^*$). We consider a standard error criterion for our comparisons. However, since the rank correlation coefficients suffer from the tied problem  that results from the bootstrap technique, we use existing  modified formulas  for some rank correlation coefficients, otherwise, the  randomization tied-treatment is applied