Asymptotic expansion of the sample correlation coefficient under nonnormality

The 2-term Edgeworth expansion, or the expansion up to order 1/n1/n, of the distribution of the sample correlation coefficient in nonnormal observations is obtained. For the expansion, the formula of the fourth cumulant of the function of sample variances and covariances of the associated observable variables is given. From a simulation, the partially weighted 2-term Edgeworth expansion is found to give smaller errors than those by the single-term or fully weighted 2-term Edgeworth expansions.