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I have read several times across different sources now that the definition of Cook's Distance, which is $$D_i=\frac{\sum_{j=1}^n(\hat{y}_j-\hat{y}_{j(i)})^2}{ps^2}$$ (where $\hat{y}_j$ is the jth fitted response value, $\hat{y}_{j(i)}$ the jth fitted response value, where the fit does not include observation i, $p$ the number of coefficients in the model, $s^2$ the MSE) is algebraically equivalent to the expression $$\frac{r_i^2}{ps^2}\frac{h_{ii}}{(1-h_{ii})^2}$$ For $r_i$ the ith residual, $h_{ii}$ the ith leverage value.
Despite numerous sources stating this (lecture notes from different universities, wikipedia, Matlab's website, etc.), I have not been able to find a single proof of this. I would very much appreciate it if someone could direct me to one such proof. It seems like a completely non-trivial equivalence to me, and I am unsure why it's so difficult to find a proof of this.

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I have found a source written by Cook himself (here is a pdf link I found):

Cook, R.D., (1977). Detection of Influential Observation in Linear Regression. Technometrics, 19(1), 15-18.

The derivation ends at equation 7 in that paper.

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  • $\begingroup$ Thanks. I am curious, how did you find it? I've been looking for a while without finding this. $\endgroup$ Dec 14, 2019 at 9:40
  • $\begingroup$ It was a long journey through the internet that I can't retrace, but someone mentioned a paper by Cook in 1979 as a reference for Cook's distance. In that paper, Cook references this 1977 paper as the introduction of (what we now call) Cook's distance. By luck he also derives the algebraic equivalent there. $\endgroup$ Dec 15, 2019 at 0:03

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