A Tutorial on Quantile Regression, Quantile Random Forests, and Quantile GBM

Have you been asked to provide prediction intervals beside the mean predictions? Prediction intervals have many use cases because they provide the range of the predicted values to give better guidance. In financial risk management, the prediction intervals for the high range can help risk managers to mitigate risks. In science, a predicted life of a battery between 100 to 110 hours can inform users when to take action. Please comment on which of the following is more applicable.

• The expected average financial loss is $40M, or
• We have 95% confidence that the financial loss will be between $10M and $70M, with an average of around $40M. Further, we have 68% confidence that the financial loss will be between $30M and $50M.

The second one, right? The first one is the prediction of an Ordinary Least Square (OLS) and the second one is a Quantile Regression (OR). For this reason, QR has received increasing attention and applied to many areas such as investment, finance, economics, medicine, and engineering.

An OLS only predicts the conditional mean Y = E[Y|X]+e. If OLS models the conditional means, why don’t we model the conditional median or any other percentiles (the term quantile in QR is the same as a percentile)? It is interesting to know…