Bounded influence estimation for regression and scale

Abstract

Bednarski and Müller [Optimal bounded influence regression and scale M-estimators in the context of experimental design, Statistics 35 (2001), pp. 349–369] introduced a class of bounded influence M estimates for the simultaneous estimation of regression and scale in the linear model with normal errors by solving the corresponding normal location and scale problem at each design point. This limits the proposal to regressor distributions with finite support. Based on their approach, we propose a slightly extended class of M estimates that is not restricted to finite support and is numerically easier to handle. Moreover, we employ the even more general class of asymptotically linear (AL) estimators which, in addition, is not restricted to normal errors. The superiority of AL estimates is demonstrated by numerical comparisons of the maximum asymptotic mean-squared error over infinitesimal contamination neighbourhoods.

Keywords:

• regression and scale
• influence curves
• asymptotically linear estimators
• M estimates
• minmax MSE over contamination neighbourhoods

Mathematics Subject Classification (2010) :

• 62F12
• 62F35
• 62G35
• 62J05

Acknowledgements

I thank Prof. Dr Helmut Rieder for his generous guidance and encouragement.

Notes

In Equation (7), we used the stochastic Landau notation of 10, i.e. in product probability as n→∞.