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A nice description of leverage in the sense that I am using it is given here so I will not repeat it.

Who originally defined leverage scores to be the diagonal elements of $X(X^TX)^{-1}X^T$?

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    $\begingroup$ This is not really the definition: it's a consequence of a definition. Belsley, Kuh, and Welsch (Regression Diagnostics, 1980) motivate it as follows. "The influence of the response value, $y_i,$ on the fit is most directly reflected in its impact on the corresponding fitted value, $\hat y_i,$ and this information is seen ... to be contained in $h_i$" (the diagonal element of the hat matrix). See BK&W pp 16 - 18 for the subsequent analysis. $\endgroup$
    – whuber
    Commented Mar 9, 2022 at 16:26
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    $\begingroup$ To the closevoter(s): I think this is perfectly on-topic here, and not opinion-based $\endgroup$ Commented Mar 9, 2022 at 17:02

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Hoaglin & Welsch (1978) say Thus we use the hat matrix to identify "high-leverage points." If this notion is to be really useful, we must make it more precise.

This suggests the term 'leverage' is not widely used at that point, and I don't have an earlier reference.

Tukey (who also coined the term 'hat matrix') uses 'leverage' in a related sense in a 1965 paper, talking about the amount of information in subsets of a sample, but not for regression. It's not used in the paper by Cook defining Cook's distance for measuring influence.

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