Asymptotic properties of Pearson׳s rank-variate correlation coefficient in bivariate normal model

• Establishment of the asymptotic closed forms of the expectation and variance of the Pearson׳s rank-variate correlation coefficient (PRVCC) under bivariate normal model.
• Investigation of the variance-stability features of Fishers z-transform on PRVCC.
• Robustness study of PRVCC for data following heavy-tailed distribution such as bivariate student distribution.

This paper establishes the asymptotic closed forms of the expectation and variance of the Pearson׳s rank-variate correlation coefficient (PRVCC) with respect to samples drawn from bivariate normal populations. The variance-stability features of Fisher׳s z-transform on PRVCC is also investigated under normal assumptions. To gain deeper insight into PRVCC, we further compare PRVCC to other two closely related correlation coefficients, namely, Pearson׳s product moment correlation coefficient (PPMCC) and Gini correlation (GC). Theoretical and simulation results reveal the advantages of PRVCC over PPMCC and/or GC in terms of mathematical tractability and smaller mean square error (MSE) under different circumstances. The newly found theoretical results along with other desirable properties enable PRVCC to be a useful alternative to the existing coefficients, especially when the samples follow distributions whose tails are heavier than that of normal distribution.