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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 1557-1570

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2009.202

\vspace{3pc} \normalsize \centerline{\large\bf OPTIMAL DESIGNS FOR
ESTIMATING THE DERIVATIVE} \vspace{2pt} \centerline{\large\bf IN
NONLINEAR REGRESSION} \vspace{.25cm}
\author{Holger Dette$^1$, Viatcheslav B. Melas$^2$ and
Petr Shpilev$^2$} \vspace{.4cm} \centerline{\it
$^1$Ruhr-Universit\"at Bochum and $^2$St. Petersburg State
University} \vspace{.55cm}  
\begin{quotation}
\noindent {\it Abstract:} We consider the problem of estimating the
derivative of the expected response in nonlinear regression models.
It is demonstrated that in many cases the optimal designs for
estimating the derivative have either on $m$ or $m-1$ support
points, where $m$ denotes the number of unknown parameters in the
model. It is also shown that the support points and weights of the
optimal designs are analytic functions, and this result is used to
construct a numerical procedure for the calculation of the optimal
designs. The results are illustrated in exponential regression and
rational regression models.\par \vspace{9pt} \noindent {\it Key
words and phrases:} Chebyshev system, $c$-optimal designs, implicit
function theorem, nonlinear regression. \par
\end{quotation}

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