Statistica Sinica 22 (2012), 655-679

doi:http://dx.doi.org/10.5705/ss.2010.155

ON THE DERIVATIVES OF THE TRIMMED MEAN

Subhra Sankar Dhar and Probal Chaudhuri

Indian Statistical Institute

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.

Key words and phrases: Contamination model, Karhunen-Loeve expansion, Pitman efficacy, proportion of contamination, test of symmetry, weak convergence of processes.