UPPER BOUNDS FOR THE L_{1}-RISK

OF THE MINIMUM L_{1}-DISTANCE REGRESSION ESTIMATOR

L. GAJEK AND M. KAŁUSZKA

Institute of Mathematics, Polish Academy of Sciences, Institute of Mathematics, Technical University of Lódz, Al. Politechniki 11, 90-924 Lódz, Poland

(Received April 30, 1991; revised February 27, 1992)

Abstract. A new estimator of a regression function is introduced via minimizing the L_{1}-distance between some empirical function and its theoretical counterpart plus penalty for the roughness. The L_{1}-risk of the estimator is bounded from above for every sample size no matter what the dependence structure of the observed random variables is. In the case of independent errors of measurement with a common variance the estimator is shown to achieve the optimal L_{1}-rate of convergence within the class of m-times differentiable functions with bounded derivatives.

Key words and phrases: Nonlinear regression, minimum distance estimation, rates of convergence.