Looking for a name for a "mean-influencing" statistic 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


*

*This specific measure has been used elsewhere -- especially in the statistical literature.

*This specific measure has a name


Thanks in advance for your time.
 A: 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
$$\eqalign{
DFBETA_i &= \frac{x_i - \bar{x}}{n - 1} = \frac{1}{n-1}\left(x_i - \frac{1}{n}\sum_{j=1}^n x_j \right) \\
&= \frac{1}{n}\sum_{j=1}^n x_j - \frac{1}{n-1}\sum_{j \ne i} x_j \text{.}
}$$
A: 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.
