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I have been using a statistic to measure "influence" or some analogous concept in the following way:

First calculate the mean for a given sample, then calculate the mean for the given sample excluding each observation in turn. Subtract the latter from the former. The result is a vector measuring the amount by which the sample mean is increased by the inclusion of each observation.

Something like this (more or less): $$ \frac{\sum_{i = 1}^{n}}{n} - \frac{\sum_{i \neq 1}^{n}}{n - 1} $$

To me, this recalls the concept of "leverage" in regression, but I am wondering whether

  1. This specific measure has been used elsewhere -- especially in the statistical literature.
  2. This specific measure has a name

Thanks in advance for your time.

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The mean is the coefficient in the regression of the data against the constant $1$. Your statistic, in this regression context, is the simplest possible example of the DFBETA diagnostic defined in Belsley, Kuh, & Welsch, Regression Diagnostics (J Wiley & Sons, 1980):

...we look first at the change in the estimated regression coefficients that would occur if the $i^\text{th}$ row were deleted. Denoting the coefficients estimated with the $i^\text{th}$ row deleted by $\mathbf{b}(i)$, this change is easily computed from the formula

$$DFBETA_i = \mathbf{b} - \mathbf{b}(i) = \frac{(X^T X)^{-1} x_i^T e_i}{1 - h_i}$$

where

$$ h_i = x_i (X^T X)^{-1} x_i^T \ldots$$

[pp 12-13, formulas (2.1) and (2.2)].

In this case the design matrix $X$ is the $n$ by $1$ matrix of ones, whence $(X^T X)^{-1} = 1/n$. The numbers $e_i$ are the residuals,

$$e_i = x_i - \bar{x}.$$

Therefore

$$\eqalign{ DFBETA_i &= \frac{x_i - \bar{x}}{n - 1} = \frac{1}{n-1}\left(x_i - \frac{1}{n}\sum_{j=1}^n x_j \right) \\ &= \frac{1}{n}\sum_{j=1}^n x_j - \frac{1}{n-1}\sum_{j \ne i} x_j \text{.} }$$

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  • $\begingroup$ This is a very useful way to think about it, and DFBETA looks exactly like what I am using. Thanks for the quick reply and great explanation. $\endgroup$ – rapidadverbssuck Jun 21 '11 at 13:09
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It's close to (if not exactly) local influence and/or Cook's distance.

JRSS B, Vol. 48, No. 2, 1986, p.133-169 is the classic paper. A bit dense but a place to start looking in the literature.

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  • $\begingroup$ Cook's distance sounds about right, thanks for the reference. $\endgroup$ – rapidadverbssuck Jun 21 '11 at 13:08

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