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Why the sample covariance estimator is unbiased, but the sample Pearson correlation coefficient is not?

I'm a bit confused because the sample Pearson coefficent was built using the sample covariance estimator and the sample variance estimator, that are both unbiased.

Thanks in advance!

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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!

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    $\begingroup$ The Wikipedia article on Jensen’s inequality has some pictures that might give some intuition. $\endgroup$
    – Dave
    Commented Apr 28, 2021 at 10:16

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