# What is this measure of error called?

I am investigating prediction errors in a context where the errors can be extremely large. Someone advised me that in addition to reporting the mean or median of the absolute errors $$|\hat{x} - x|$$ that I should also report the mean or median of this quantity: $$\exp(|\ln(\hat{x}) - \ln(x)|)$$ ...the idea being that, for example, an error of twice the true value is treated as being of equivalent magnitude as an error of half the true value. This person referred to this as the "relative error," but Googling "relative error" suggests that this refers to something else. Is there a name for the measure I was advised to report? Or is there a specific subliterature in which this measure is referred as "relative error" (and if so, what subliterature is this, and could I be pointed to an example paper)?

• For an account of relative errors and various ways to define them, see stats.stackexchange.com/a/201864/919. The quantity you ask about is a simple variation of the $d_\infty$ measure in the Wikipedia article. – whuber Jan 22 at 16:04

## 1 Answer

@whuber's comment gave me the direction I needed to find additional relevant information on this topic. It seems there are many approaches to calculating relative error. The particular measure I inquired about is the exponential of the absolute value of what has been referred to as the "log change" or the "log difference", and there is a spirited defense of it here:

Törnqvist, L., Vartia, P., & Vartia, Y. O. (1985). How should relative changes be measured? The American Statistician, 39(1), 43-46.