Interval estimation for the normal correlation coefficient

Inference concerning the correlation coefficient of two random variables from the bivariate normal distribution has been investigated by many authors. In particular, Fisher [1915. Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika 10, 507-521] and Hotelling [1953. New light on the correlation coefficient and its transform. J. Roy. Statist. Soc. Ser. B 15, 193-232] derived various exact forms of the density for the sample correlation coefficient. However, obtaining confidence intervals based on these densities can be computational intensive. Fisher [1921. On the "probable error" of a coefficient of correlation deduced from a small sample. Metron 1, 3-32], Hotelling [1953. New light on the correlation coefficient and its transform. J. Roy. Statist. Soc. Ser. B 15, 193-232], and Ruben [1966. Some new results on the distribution of the sample correlation coefficient. J. Roy. Statist. Soc. Ser. B 28, 513-525] suggested several simple approximations for obtaining confidence intervals for the correlation coefficient. In this paper, a likelihood-based higher-order asymptotic method is proposed to obtain confidence intervals for the correlation coefficient. The proposed method is based on the results in Fraser and Reid [1995. Ancillaries and third order significance. Utilitas Math. 7, 33-53] and Fraser et al. [1999. A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika 86, 249-264]. Simulation results indicated that the proposed method is very accurate even when the sample size is small.