  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.
