Fused LASSO as Non-Crossing Quantile Regression

Quantile crossing has been an ever-present thorn in the side of quantile regression. This has spurred research into obtaining densities and coefficients that obey the quantile monotonicity property. While important contributions, these papers do not provide insight into how exactly these constraints influence the estimated coefficients. This paper extends non-crossing constraints and shows that by varying a single hyperparameter ($\alpha$) one can obtain commonly used quantile estimators. Namely, we obtain the quantile regression estimator of Koenker and Bassett (1978) when $\alpha=0$, the non crossing quantile regression estimator of Bondell et al. (2010) when $\alpha=1$, and the composite quantile regression estimator of Koenker (1984) and Zou and Yuan (2008) when $\alpha\rightarrow\infty$. As such, we show that non-crossing constraints are simply a special type of fused-shrinkage.