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\footnotesize
\noindent Statistica Sinica {\bf 17}(2007), 713-724

\vspace{3pc} \normalsize
\centerline{\large\bf A TEST FOR
CONSTANCY OF ISOTONIC REGRESSIONS }
\vspace{2pt}
\centerline{\large\bf USING THE $\mbf{L_2}$-NORM}
\vspace{.25cm}
\author{Ana Colubi, J. Santos Dom\'{\i}nguez-Menchero and Gil
Gonz\'{a}lez-Rodr\'{\i}guez}
\vspace{.4cm}
\centerline{\it Universidad de Oviedo}
\vspace{.55cm}
\begin{quotation}
\noindent {\it Abstract:}
In this paper we present a nonparametric
procedure for testing the constancy of an isotonic regression. We
introduce a family of statistics based on the  $L_2$-norm of the
difference of an isotonic estimate of the regression and the
estimate under the (null) hypothesis that it is a constant. We
propose to choose as isotonic estimate a tail-smoothed version of
the usual least squares isotonic regression estimate. We write
 the selected statistic in terms of  a certain functional in order to analyze its asymptotic distribution from the continuity
 properties of the functional. Finally, we show some simulations to compare  the
stated procedure with the well-known parametric
$\overline{\chi}^2_{01}$.\par \vspace{9pt} \noindent {\it Key words
and phrases:} Isotonic regression, $L_2$-norm, testing constant
regression. \par
\end{quotation}

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