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Suppose we have a dataset $(x_i, y_i)$ where both $x$ and $y$ are monotonically non-decreasing with $i$. So obviously the Spearman rank correlation between $x$ and $y$ is 1.

However, what is the smallest possible Pearson correlation between $x$ and $y$?

I wasn't able to find an explicit theorem about this. So I'm wondering what is the lower bound and how to prove it.

jkff
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  • Zero is attained by datasets of the form $(x_i, 0).$ I challenge you to achieve a negative Pearson correlation with co-monotonic data! (Since it will be proportional to the covariance, consider computing the covariance in the way I described at https://stats.stackexchange.com/a/18200/919.) – whuber Dec 11 '18 at 22:47
  • Is correlation defined for datasets of the form $(x_i, 0)$? The covariance is certainly zero, but so is the standard deviation of $y$ in the denominator. – jkff Dec 11 '18 at 23:19
  • That's right, but in such cases one might revert to the relationship between correlation coefficient and least-squares slope to establish a reasonable convention that such datasets have zero correlation. Otherwise, you need to contemplate arbitrarily large datasets like $(1,0),(2,0),\ldots,(n-1,0), (n, 1)$ and consider the infimum of their correlations. – whuber Dec 11 '18 at 23:27

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