Abstract. We consider a fixed-design regression model with long-range dependent errors which form a moving average process. Taking into account different behavior of regression estimators in such a model and in a random-design regression model discussed in Csörgő and Mielniczuk [5], we introduce an artificial randomization of grid points at which observations are taken in order to diminish the impact of strong dependence of errors. The resulting estimator is shown to exhibit smoothing dichotomy with the variance in both cases tending to $0$ more quickly than in the fixed design case. Moreover, we establish a uniform convergence rate of the regression function estimators which also reflects the dichotomous behaviour of the regression estimator. Simulation results indicate significant improvement for moderate sample sizes when randomization is employed.

AMS Subject Classification (1991): 62G07, 60F17

Keyword(s): Long- and short-range dependence, randomization, kernel estimator, linear process, random- and fixed-design regression, smoothing dichotomy

Received January 22, 2007, and in final form October 25, 2007. (Registered under 6463/2009.)