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Spearman's Rho (Spearman's Rank Correlation Coefficient) is a measure of nonlinear dependence between two random variables. If random variables
and
have joint distribution
_{ }
and random vectors
_{ }
_{ }
and
_{ }
are independent realizations from that distribution, then Spearman's rho of
and
equals

If and have continuous marginal distributions then

Still, no single generalization has been widely accepted.

Note that definition (1) depends on ranks only. We only care if

Spearman's rho has direct relation to the copula function

The copula function does not depend on marginal distributions and captures what happens to and if they are transformed into random variables uniformly distributed on [0,1]. The formula above signals once again that Spearman's rho does not depend on marginal distributions of and and is invariant to any monotonically increasing transformations of and

When the joint distribution of and is unknown Spearman's rho can be estimated from the data as the correlation of ranks. Let

The estimator of Spearman's rho is given by

Identical values are assigned the same fractional rank, which is equal to the average of their positions in the ascending order of the values. For that reason the estimator is suitable for both discrete and continuous distributions. As the sample size converges to infinity, the estimator converges to the true Spearman's rho and can be used to test if the true Spearman's rho equals 0.

The sample rank correlation coefficient is a

Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed). New York: Springer.

Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Extremes in Nature: An Approach Using Copulas. Springer.

Corder, G.W., & Foreman, D.I. (2014). Nonparametric Statistics: A Step-by-Step Approach. Wiley, Hoboken, New Jersey.

Gibbons, J. D., & Chakraborti, S. (2003). Nonparametric Statistical Inference (4th ed). New York: Marcel Dekker.

Nešlehová, J. (2007). On Rank Correlation Measures for Non-continuous Random Variables. Journal of Multivariate Analysis, Vol. 98, Issue 3, pp. 544-567.

Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, Vol. 15, pp. 72–101.

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