Quantile estimators with orthogonal pinball loss function

To guarantee stable quantile estimations even for noisy data, a novel loss function and novel quantile estimators are developed, by introducing the effective concept of orthogonal loss considering the noise in both response and explanatory variables. In particular, the pinball loss used in classical quantile estimators is improved into novel orthogonal pinball loss (OPL) by replacing vertical loss by orthogonal loss. Accordingly, linear quantile regression (QR) and support vector machine quantile regression (SVMQR) can be respectively extended into novel OPL†based QR and OPL†based SVMQR models. The empirical study on 10 publicly available datasets statistically verifies the superiority of the two OPL†based models over their respective original forms in terms of prediction accuracy and quantile property, especially for extreme quantiles. Furthermore, the novel OPL†based SVMQR model with both OPL and artificial intelligence (AI) outperforms all benchmark models, which can be used as a promising quantile estimator, especially for noisy data.