I've been looking for an expression for the expected value and variance of the sample correlation coefficient. Most of the sources I've found say
$$
Var(Cor(X, Y)) \approx \frac{(1-\rho^2)^2}{n-1},
$$
as the variance of the sample correlation coefficient, but this assumes that $X$ and $Y$ follow a bivariate normal distribution.
There also seems to be several approaches to series expansion of the function to approximate the moments of the correlation function. However, it has not been clear to me what the assumptions are (e.g., normality), nor which one is the most updated expression.
So, does anyone know of an expression (approximate or not) for the expected value and variance of the correlation coefficient (Pearsons) that does not assume a particular distribution on the random variables?
Update:
Some of my sources:
Assumes bivariate normal distribution:
Published works:
Hotelling (1953): New Light on the Correlation Coefficient and its Transforms. (http://www.jstor.org/stable/2983768)
Fisher (1921): (https://digital.library.adelaide.edu.au/dspace/bitstream/2440/15169/1/14.pdf)
Web sources:
Gerstman (http://www.sjsu.edu/faculty/gerstman/StatPrimer/correlation.pdf)
Stack Exchange (Standard error from correlation coefficient)
Wikipedia (https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient#Inference)
Holland (http://strata.uga.edu/6370/lecturenotes/correlation.html)
Doesn't state the assumption of bivariate normality, but it should be assumed:
Stack Overflow (https://stackoverflow.com/questions/16097453/how-to-compute-p-value-and-standard-error-from-correlation-analysis-of-rs-cor)
I don't understand this one, unfortunately, but it seems it would be a fruitful approach:
Hawkings (1989) – Using U Statistics to Derive the Asymptotic Distributino of Fischer's Z Statistic (http://www.jstor.org/stable/2685369)
Best Answer
I cannot give you one expression, but here are several articles that cover some non-normal cases:
Browne, M. W., & Shapiro, A. (1986). The asymptotic covariance matrix of sample correlation coefficients under general conditions. Linear Algebra and its Applications, 82, 169-176.
Gayen, A. K. (1951). The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes. Biometrika, 38, 219-247.
Kowalski, C. (1972). On the effects of non-normality on the distribution of the sample product-moment correlation coefficient. Applied Statistics, 21, 1-12.
Subrahmaniam, K., & Gajjar, A. V. (1980). Robustness to nonnormality of some transformations of the sample correlation coefficient. Journal of Multivariate Analysis, 10, 60-77.
Yuan, K.-H., & Bentler, P. M. (2000). Inferences on correlation coefficients in some classes of nonnormal distributions. Journal of Multivariate Analysis, 72, 230-248.