# Better to Minimize Absolute Error or Sum of Squared Error?

I have an Excel model which predicts the number of customers for a given month. The prediction depends on a churn rate. I have the absolute error (actual vs predicted), along with squared error and sum of square error.

My question is:

Would it better to find a churn rate that minimizes the absolute for each period (year, month) or find a churn rate that minimizes the sum of squared errors? Does the former even make sense to do? • This depends on your loss function, I think. – Richard Hardy Nov 28 '18 at 19:07

## 1 Answer

If you minimise absolute error, you are implicitly assuming that your errors are Laplacian distributed and if you minimise mean-squared error, you're implicitly assuming they're normally distributed.

Due you have any reason a-priori to believe one of the other? Are your tails long? Pedantically, users are discrete, so are either appropriate ? If the numbers are quite large, continuous isn't a bad approximation but if your numbers are small, you might consider a discrete distribution leading to an entirely different distribution.

A-priori arguments aside, train a model using least squares, and then plot a histogram of your prediction errors $$y_{i} - \hat{y}_{i}$$. Does this distribution look normal? Train the model using absolute error and plot $$y_{i} - \hat{y}_{i}$$, does this distribution look Laplacian?

• The conclusions you draw about "implicitly" themselves make the implicit assumption that you are using maximum likelihood. As @Richard Hardy suggests in a comment to the question, it's often more appropriate to consider the loss function rather than exploring distributional assumptions. – whuber Nov 28 '18 at 20:23
• yes, that's fair. Would your default approach not be to use max-likelihood or max posterior unless you had a good reason to use something else? – gazza89 Nov 28 '18 at 20:33
• My default is to explore the loss with the client in an effort to gather information relevant to deciding on an appropriate procedure. – whuber Nov 28 '18 at 20:35