Robustness of the Simultaneous Estimators of Location and Scale From Approximating a Histogram by a Normal Density Curve

Abstract

The robust properties of the simultaneous estimators of location and scale parameters (μ*, σ*) proposed by Brown and Hwang are studied. As a pair of simultaneous M estimators of location and scale, their asymptotic efficiencies (0.650 for μ* and 0.541 for σ*) are higher than those for median (0.637) and median absolute deviation (0.368) under the normal distribution. Simulation indicates that the distributions of and are much flatter than those based on the sample mean and the sample standard deviation under the normal distribution when the sample size is small.

KEY WORDS::

• Asymptotic efficiency
• Gross-error sensitivity
• Robust estimation

Acknowledgments

This work was partially supported by National Science Foundation (NSF) grants DMS-0603761, DMS-0904125, and Mathematical Sciences Center at Tsinghua University, Beijing, China. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the view of the NSF. We are greatly indebted to the editor, the associate editor, and the referees whose constructive suggestions and comments helped us to prepare a far better version of the original submission.