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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 655-679

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2010.155

\vspace{3pc} \normalsize \centerline{\large\bf ON THE DERIVATIVES OF
THE TRIMMED MEAN} \vspace{.25cm}
\author{Subhra Sankar Dhar and Probal Chaudhuri}
\vspace{.4cm} \centerline{\it Indian Statistical Institute}
\vspace{.55cm} 
\begin{quotation}
\noindent {\it Abstract:}
 The trimmed mean is well-known for being
more robust and for having better mean square error than the mean
when data arise from non-Gaussian distributions with heavy tails. In
this paper, we consider the derivatives of the trimmed mean with
respect to the trimming proportion, and investigate some statistical
applications of those derivatives. We develop a diagnostic tool
based on the first derivative of the trimmed mean to determine
whether the data is generated from a symmetric distribution or not.
We also propose a test of symmetry of the distribution based on the
first derivative, and demonstrate by theoretical and simulation
studies that it performs better than several other well-known tests
of symmetry. Further we introduce an estimate, based on the second
derivative of the trimmed mean, of the contamination proportion
$\beta\in(0, 1/2)$ in the contamination model $F(x) = (1 -
\beta)H(x) + \beta G(x)$, where $H$ and $G$ are two distributions
such that $G$ is stochastically larger than $H$. In addition to some
theoretical studies, we carry out a detailed numerical study to show
that, in many situations, our proposed estimate of the contamination
proportion outperforms other estimates that are based on the idea of
maximum likelihood estimation in mixture models.\par \vspace{9pt}
\noindent {\it Key words and phrases:} Contamination model,
Karhunen-Loeve expansion, Pitman efficacy, proportion of
contamination, test of symmetry, weak convergence of processes.
\par
\end{quotation}
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