Amazingly, transforming an unbiased estimator often results in a biased estimator. This is how the sample standard deviation is a biased estimator, despite the sample variance being unbiased. This fact comes from something called Jensen’s inequality. For a concave function $f$, such as a square root:

$$ (\mathbb{E}[X])\ge \mathbb{E}[f(X)] $$

Equality holds if and only if $f$ is a straight line (or if $X$ is constant).

So why is $\hat{\rho}=\dfrac{\widehat{cov}(X,Y)}{s_X s_Y}$ biased? The standard deviation estimators are biased!

Amazingly, transforming an unbiased estimator often results in a biased estimator. This is how the sample standard deviation is a biased estimator, despite the sample variance being unbiased. This fact comes from something called Jensen’s inequality. For a concave function $f$, such as a square root:

$$ f(\mathbb{E}[X])\ge \mathbb{E}[f(X)] $$

Equality holds if and only if $f$ is a straight line (or if $X$ is constant).

So why is $\hat{\rho}=\dfrac{\widehat{cov}(X,Y)}{s_X s_Y}$ biased? The standard deviation estimators are biased!