On Automatic Boundary Corrections*

Many popular curve estimators based on smoothing have diicul-ties caused by boundary eeects. These eeects are visually disturbing in practice and can play a dominant role in theoretical analysis. Local polynomial regression smoothers are known to correct boundary eeects automatically. Some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections. In both settings of density and regression estimation, we investigate best weight functions for local polynomial tting at the endpoints and nd a simple solution. The solution is universal for general degree of local polynomial tting and general order of estimated derivative. Furthermore, such local polynomial estimators are best among all linear estimators in a weak minimax sense. And they are highly eecient even in the usual linear minimax sense.