Suppose $r$ is the sample Pearson correlation coefficient estimator of two random variables $X$ and $Y$, while $rho$ is the population correlation coefficient. From various sources, e.g., Is the sample correlation coefficient an unbiased estimator of the population correlation coefficient?, we know that, if $X$ and $Y$ obey the bi-variate Gaussian random variables with population correlation coefficient $\rho$, the bias of the sample estimator is $$ E[r]=\rho\left[ 1-\dfrac{1-\rho^2}{2N}+O(1/N^2) \right]. $$ Then what is the sampling bias of the estimator $r^2$?
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