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The problem with sample correlation estimator defined as: $$r_{sample} =\frac{\sum\left(X_i - \bar{X}\right)\left(Y_i - \bar{Y}\right)}{\sqrt{\sum\left(X_i-\bar{X}\right)^2\left(Y_i-\bar{Y}\right)^2}}.$$ is that it is biased. The bias is in fact downward i.e. $r_{sample}$ tends to be lower than population $\rho$. Therefore when we average biased estimator we ...


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