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\footnotesize
\noindent Statistica Sinica {\bf 12}(2002), 965-990
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\centerline{\large\bf UNIMODAL DENSITY ESTIMATION}
\vspace{2pt}
\centerline{\large\bf USING KERNEL METHODS}
\vspace{.25cm}

\centerline{Peter Hall$^1$ and Li-Shan Huang$^{1,2}$}
\vspace{.4cm}
\centerline{\it $^1$Australian National University and
$^2$University of Rochester}
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\begin{quotation}
\noindent {\it Abstract:}
We suggest a method for rendering a
standard kernel density estimator unimodal:  tilting the empirical
distribution.  It is proposed that the amount of tilting be chosen
in order to minimise, subject to unimodality, the integrated squared
distance between a conventional density estimator and its tilted
version.  This approach has an interesting data-compression aspect,
in that the algorithm often implicitly summarises the dataset in a
relatively small subsample as part of the process of enforcing
unimodality.  Another feature is that, no matter what the chosen
bandwidth, the algorithm produces (with probability~1, for each
sample size) a proper density estimate.  Thus, it may be employed as
an adjunct to any of the many popular bandwidth selection rules for
density estimation.  We show theoretically that in classes of
densities that are of practical interest, the method enhances
performance without suffering any deleterious first-order impact on
asymptotic accuracy, for example as reflected in integrated squared
error.  In such cases, and except in places where the true density
is virtually flat, the constrained density estimate is first-order
equivalent to its unconstrained counterpart.  The case where the
number of modes is constrained to equal a number greater than~1 is
also considered.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Bandwidth, biased bootstrap, integrated
squared error, mode, nonparametric density estimation, order
restricted inference, power divergence, tilting, weighted bootstrap.\par
\end{quotation}
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