A remark on certain classical criteria of mathematical statistics
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.