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Root-mean-square error is frequently used in for calculating the error between a predicted value and actual value. The formula for RMSE is given below:

$\mathrm{RMSE} = \sqrt{\frac{\sum_{t=1}^{n}{(y_t - \hat{y}_t)^2}}{n}}$

My question is; why we raise the absolute error to the second power (and then calculate the square-root of the whole thing), but not something else(e.g., 3 or 4)? Is it just a convention, or there is a mathematical explanation for it? Thanks.

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You can have other powers: http://en.wikipedia.org/wiki/Power_mean (I know, Wikipedia, but this article looks pretty good). Two has been chosen since it has nice properties (e.g. the standard deviation is the RMS of the deviations, and that worked out well).

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For one thing, mean square error (MSE) is a nicely differentiable way to ensure the terms being summed are non-negative, so error accumulates rather than cancels. Using absolute value would accomplish the same thing, but you lose differentiability that way.

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Because the mean minimizes a quadratic, e.g.

$\mu$ minimizes $E(X-t)^2$

In other words, you're trying to find parameter values such that your predictions are as close to the conditional mean of $y_i$ given $x_i$ as possible (assuming regression context).

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