We consider the problem of estimating the correlation in bivariate normal data when the means and variances are assumed known, with emphasis on the small sample case. We consider eight different estimators, several of them considered here for the first time in the literature. In a simulation study, we found that Bayesian estimators using the uniform and arc-sine priors outperformed several empirical and exact or approximate maximum likelihood estimators in small samples. The arc-sine prior did better for large values of the correlation. For testing whether the correlation is zero, we found that Bayesian hypothesis tests outperformed significance tests based on the empirical and exact or approximate maximum likelihood estimators considered in small samples, but that all tests performed similarly for sample size 50. These results lead us to suggest using the posterior mean with the arc-sine prior to estimate the correlation in small samples when the variances are assumed known.

中文翻译：

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估计具有已知方差和小样本量的双变量正态数据的相关性

我们考虑在均值和方差已知时估计双变量正态数据相关性的问题，重点是小样本情况。我们考虑了八种不同的估计量，其中一些是在文献中首次考虑的。在模拟研究中，我们发现使用均匀和反正弦先验的贝叶斯估计量在小样本中优于几个经验和精确或近似的最大似然估计量。对于较大的相关值，反正弦先验效果更好。为了测试相关性是否为零，我们发现贝叶斯假设检验优于基于在小样本中考虑的经验和精确或近似最大似然估计量的显着性检验，但对于样本大小 50，所有检验的执行情况相似。