A remark on certain classical criteria of mathematical statistics

Abstract

The paper is devoted to asymptotical features of the standard statistical test for the significance of Pearson correlation between random variables x and y. Although this test (sometimes called t-test for the correlation coefficient) is only justified if the distribution of (x, y) is Gaussian, it is very widespread and is included in many statistical packages. However, in practice the Gaussian assumption usually fails. Thus, it is worth to describe the area of the correct application of t-test for big sample sizes. It is proven that under certain additional conditions, this test is asymptotically precise if x and y are independent, while the simple lack of correlation is not sufficient for such a feature. Also, the general asymptotically precise and consistent test for the significance of correlation coefficient is proposed. Computational experiments show that this test can be successfully applied in practice. Under modifications these results are transferred to the partial correlation coefficient.