In this paper, we consider the problem of inference on the correlation coefficient from a bivariate normal distribution with known variances. The problem is theoretically interesting because the model belongs to the curved exponential family and the standard inferential methods cannot be directly applied. It is also of practical interest because the problem arises naturally in many applications with small sample sizes, and accurate inference for the correlation coefficient is essential in those studies. The modified signed log likelihood ratio test is proposed which has a known third order rate of convergence. Simulation studies are performed to compare the accuracy of the proposed method with   some of the existing Bayesian methods and the results show that the proposed method has remarkable accuracy even when the sample size is extremely small.

ancillary direction, curved exponential family, modified signed log likelihood ratio statistic, p-value function, standardized maximum likelihood estimate departure.