What are the implicit assumptions that justify the usage of the “relative error”?

Question

In applied statistics, for example, analyzing data from a science experiment, we sometimes use "absolute error" while at the most of time, we calculate the "relative error". Generally, under what circumstances should we use the "relative error" instead of "the absolute error"?

If we use relative error, are we basically implicitly assuming that the variance of the sample distribution is positively correlated with the value of the mean? Is there a fundamental axiom in mathematics or statistics called "scale independence"?

It is similar to the financial math literature, the variance of the random variable $X_t$ is usually correlated with the exact value of $X_{t−1}$. If $X_{t−1}$ is greater, the variance of $X_t$ is greater. This often leads to something like a Lognormal distribution.

Answer 1

I think you are getting a bit ahead of yourself. Both absolute error and relative error are measures of error that are useful for different purposes. There are no assumptions required to use either of these measures$^\dagger$, so long as they are correctly interpreted. Of course, the statistical properties of these measures will depend on the underlying distribution of the values being predicted and the predictions being made, so if you want to assert a particular distributional property of one of these measures, you may need to make some assumptions. So, if you are asking what assumptions are needed, then naturally you first need to ask, for what? What statistical property are you seeking?

If you have a prediction $\hat{x}_t$ of an unknown value $x_t$ then the absolute error $\varepsilon_t = |\hat{x}_t - x_t|$ gives a measure of the amount by which you are wrong in your prediction. Often it is the case that this is not particularly useful on its own, since this amount might be large or small in comparison to the size of the thing you are predicting. For this reason, analysts often prefer to measure their error using the relative error $\eta_t = |\hat{x}_t - x_t| / |x_t|$.

If you are seeking a particular statistical property for one of these measures, then please specify the property you would like it to have, and then we can help to suggest sufficient conditions that would yield the required property.

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$^\dagger$ Strictly speaking, you need to have $x_t \neq 0$ to calculate the relative error, so this could possibly be taken as a required assumption. Alternatively you could merely adopt the convention that $\eta_t = \infty$ when $x_t = 0$.