Robustness of squared Msplit(q) estimation: Empirical analyses

Abstract

This paper concerns squared M_split(q) estimation and its robustness against outliers. Previous studies in this field have been based on theoretical approaches. It has been proven that a conventional analysis of robustness is insufficient for M_split(q) estimation. This is due to the split of the functional model into q competitive ones and, hence, the estimation of q competitive versions of the parameters of such models. Thus, we should consider robustness from the global point of view (traditional approach) and from the local point of view (robustness in relation between two “neighboring” estimates of the parameters). Theoretical considerations have generally produced many interesting findings about the robustness of M_split(q) estimation and the robustness of the squared M_split(q) estimation, although some of features are asymptotic. Therefore, this paper is focused on empirical analysis of the robustness of the squared M_split(q) estimation for finite samples and, hence, it produces information on robustness from a more practical point of view. Mostly, the analyses are based on Monte Carlo simulations. A different number of observation aggregations are considered to determine how the assumption of different values of q influence the estimation results. The analysis shows that local robustness (empirical local breakdown points) is fully compatible with the theoretical derivations. Global robustness is highly dependent on the correct assumption regarding q. If it suits reality, i.e. if we predict the number of observation aggregations and the number of outliers correctly, then the squared M_split(q) estimation can be an alternative to classical robust estimations. This is confirmed by empirical comparisons between the method in question and the robust M-estimation (the Huber method). On the other hand, if the assumed value of q is incorrect, then the squared M_split(q) estimation usually breaks down.