ASYMPTOTIC MINIMAX PROPERTIES OF L-ESTIMATORS OF SCALE

This paper asks whether or not the efficient L-estimator of scale corresponding to the least informative distribution in ε-contamination and Kol-mogorov neighbourhoods of certain distributions possesses the saddlepoint property. This is of interest since the saddlepoint property implies the mini-max property, namely, that the supremum of the relative asymptotic variance of an L-estimator is minimized by the efficient estimator corresponding to that member of the distributional class with minimum Fisher information for scale. Our findings are negative in all cases investigated.     

The asymptotic behaviour of a class of L.-estimators under long-range dependence

This paper obtains asymptotic representations of a class of L-estimators in a linear regression model when the errors are a function of long-range-dependent Gaussian random variables. These representations are then used to address some of the efficiency robustness properties of L-estimators compared to the least-squares estimator. It is observed that under the Gaussian error distribution, each member of the class has the same asymptotic efficiency as that of the least-squares estimator. The results are obtained as a consequence of the asymptotic uniform linearity of some weighted empirical processes based on long-range-dependent random variables.     

The total bootstrap median: a robust and efficient estimator of location and scale for small samples

We propose the total bootstrap median (TBM) as a robust and efficient estimator of location and scale for small samples. We demonstrate its performance by estimating the mean and variance of a variety of distributions. We also show that, if the underlying distribution is unknown and there is either no contamination or low to moderate contamination, the TBM provides a better estimate of the mean, in mean square terms, than the sample mean or the sample median. In addition, the TBM is a better estimator of the variance of the underlying distribution than the sample variance or the square of the bias-corrected median absolute deviation from the median estimator. We also show that the TBM is an explicit L-estimator, which allows a direct study of its properties.