I am doing some research on the Spearman Rank Correlation Coefficient; all the references I can find refer essentially to a sample statistic. That is, given a sample of the jointly distributed $(x_i,y_i)$, one can compute the Spearman Coefficient between $x$ and $y$; I am wondering if there is a population equivalent. My guess would be that it is defined as $$E[sign((x_i - x_j)(y_i - y_j))],$$ where $(x_i,y_i)$ and $(x_j,y_j)$ are i.i.d draws from the joint distribution. My question:
- is there a widely accepted definition of the population Spearman? (references?)
- does it match my intuition?
- is the sample Spearman an unbiased estimator of the population Spearman?