Derivation of the Exact Moments of the Distribution of Pearsons Correlation over Permutations of Data
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Pearson's correlation is one of the most widely used measures of association today, the importance of which to modern science cannot be understated. Two of the most common methods for computing the p-value for a hypothesis test of this correlation method are a t-statistic and permutation sampling. When a dataset comes from a bivariate normal distribution under specific data transformations a t-statistic is exact. However, for datasets which do not follow this stipulation, both approaches are merely estimations of the distribution of over permutations of data. In this paper we explicitly show the dependency of the permutation distribution of Pearson's correlation on the central moments of the data and derive an inductive formula which allows the computation of these exact moments. This has direct implications for computing the p-value for general datasets which could lead to more computationally accurate methods.
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