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%----------------------------------------------------------------------

\baselineskip 16pt
\centerline{\twlss
       ASYMPTOTICALLY OPTIMAL TESTS AND OPTIMAL DESIGNS}
\centerline{\twlss
       FOR TESTING THE MEAN IN REGRESSION MODELS WITH}
\centerline{\twlss
       APPLICATIONS TO CHANGE-POINT PROBLEMS}
\baselineskip 12pt
\vskip 11pt
\centerline{\tenss W{\egtss OLFGANG} B{\egtss ISCHOFF} {\egtss AND} 
F{\egtss
RANK} M{\egtss
ILLER}}
\baselineskip 11pt
\vskip 10pt
\centerline{\egtit Institute of Mathematical Stochastics, Department 
of
Mathematics,
University of Karlsruhe,}
\centerline{\egtit D-76128 Karlsruhe, Germany}
\vskip 9pt
\centerline{\egtrm (Received July 27, 1998;
       revised July 2, 1999)}
\vskip 19pt
%
%-----------------------------------------------------------------
% Abstract5
%-----------------------------------------------------------------
%
{\ninm\ninrm\narrower
\baselineskip 11pt
\parindent 0pt
{\ninbf Abstract.}\m Let a linear regression model be given with an
experimental
region $[a,b] \subseteq \bf R$ and regression functions
$f_1,\ldots,f_{d+1}: [a,b] \to \bf R$. In practice it is an important
question whether a certain regression function $f_{d+1}$, say, does 
or does
not belong to
the model. Therefore, we investigate the test problem
$ H_0: $ ``$f_{d+1}$ does not belong to the model" against $K: $ ``$f_{d+1}$
belongs
to the model" based on the least-squares residuals of the observations 
made
at design points of the experimental region $[a,b]$. By a new functional
central limit theorem
given in Bischoff (1998, {\ninit Ann.\ Statist.}, {\ninbf 26}, 
1398--1410),
we are able
to determine optimal tests in an
asymptotic way. Moreover, we introduce the problem of experimental design
for the optimal test statistics.
Further, we compare the asymptotically optimal test with the likelihood 
ratio
test (F-test) under the assumption that the error is normally distributed.
Finally, we consider real change-point
problems as examples and investigate by simulations the behavior of 
the
asymptotic test for finite sample sizes. We determine optimal
designs for these examples.

\vskip 9pt
{\ninit Key words and phrases\/}:\m Asymptotically optimal tests,
linear regression, F-test, likelihood ratio test,
Gaussian processes, optimal designs, change-point problem, quality 
control.

\par}
\baselineskip 12pt
\vskip 1pt

\end
