4
$\begingroup$

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!

$\endgroup$

1 Answer 1

6
$\begingroup$

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!

$\endgroup$
1
  • 3
    $\begingroup$ The Wikipedia article on Jensen’s inequality has some pictures that might give some intuition. $\endgroup$
    – Dave
    Apr 28, 2021 at 10:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.