Asymptotic Properties of Order Statistics Correlation Coefficient in the Normal Cases

We have previously proposed a novel order statistics correlation coefficient (OSCC), which possesses some desirable advantages when measuring linear and monotone nonlinear associations between two signals. However, the understanding of this new coefficient is far from complete. A lot of theoretical questions, such as the expressions of its distribution and moments, remain to be addressed. Motivated by this unsatisfactory situation, in this paper we prove that for samples drawn from bivariate normal populations, the distribution of OSCC is asymptotically equivalent to that of the Pearson´s product moment correlation coefficient (PPMCC). We also reveal its close relationships with the other two coefficients, namely, Gini correlation (GC) and Spearman´s rho (SR). Monte Carlo simulation results agree with the theoretical findings.