Robust Estimation of Non-linear Aspects

As a first step for dealing with efficient robust estimation in non-linear models, we regard the problem of efficient robust estimation of non-linear aspects (functions) φ(β) of the unknown parameter β of a linear model. For robust estimation of a general non-linear aspect we propose estimators which are based on one-step-M-estimators and derive their asymptotic behaviour at the contaminated linear model, where the errors have contaminated normal distributions. The asymptotic behaviour provides criteria for robustness and optimality of the estimators and the corresponding designs. Because it is impossible to find globally optimal robust estimators and designs locally optimal solutions are used for efficiency comparisons. Simple formulas for the efficiency rates are given for the general case. Using these results the efficiency rates for estimating robustly the relative variation of a circadian rhythm are calculated. These efficiency rates are very similar to those for non-robust estimation although on principle there is an important difference.

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