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A SIMPLE GOODNESS-OF-FIT TEST FOR LINEAR MODELS

UNDER A RANDOM DESIGN ASSUMPTION

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HOLGER DETTE AND AXEL MUNK

*Fakultät für Mathematik, Ruhr-Universität Bochum,*

Universitätstr. 150, D-44780 Bochum, Germany
(Received February 6, 1997; revised June 9, 1997)

**Abstract.**
Let (*X*, *Y*) denote a random vector
with decomposition *Y* = *f*(*X*) + *varepsilon* where *f*(*x*) = *E* [*Y* | *X* = *x* ]
is the regression of *Y* on *X*. In this paper we propose
a test for the hypothesis that *f* is a linear
combination of given linearly independent regression
functions *g*_{1}, ...., *g*_{d}. The test is based on an
estimator of the minimal *L*^{2}-distance between *f* and
the subspace spanned by the regression functions. More
precisely, the method is based on the estimation of
certain integrals of the regression function and
therefore does not require an explicit estimation of the
regression. For this reason the test proposed in this
paper does not depend on the subjective choice of a
smoothing parameter. Differences between the problem of
regression diagnostics in the nonrandom and random design
case are also discussed.

*Key words and phrases*:
Nonparametric
regression check, validation of goodness of fit,
*L*^{2}-distance, equivalence of regression functions,
random design.

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