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UPPER BOUNDS FOR THE *L*_{1}-RISK

OF THE MINIMUM *L*_{1}-DISTANCE REGRESSION ESTIMATOR

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L. GAJEK AND M. KALUSZKA

*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.

**Source**
( TeX ,
DVI ,
PS )