Data Analysis and Applications 1

4S-weighted Instrumental Variables

This chapter deals with two problems – with the situation when the orthogonality condition is broken and with the problem when an atypical data set contains a significant amount of information in a group of good leverage points but includes also a “troublesome” group of outliers.

Several robust methods were recently modified in order to overcome problem with the broken orthogonality condition, employing typically the idea of instrumental variables. In an analogous way, modified S-weighted estimator is also able to cope with broken orthogonality condition. We prove its consistency and we offer a small pattern of results of simulations.

It is believed that the bad leverage points are a more challenging problem in identification of underlying regression model than outliers. We show that sometimes outliers can also represent an intricate task.

4.1. Summarizing the previous relevant results

The median is the only classical statistic that is able to cope with high contamination, even 50%, and to give reasonable information about the location parameter of a data set. When Peter Bickel (Bickel 1975) opened the problem of possibility to construct an analogy of median in the framework of regression model, that is, an estimator of regression coefficients with 50% breakdown point, nobody had an idea how long and painful way to the solution we would have to go.

It seemed several times that we had achieved solution but finally always a bitter disappointment ...