A parameter can have several estimators. Among these, the "best estimators" are always assumed to have some special features that can be exploited. For an instance, the CLR model has several special assumptions, and if some of the properties are allowed to deviate from these assumptions, new estimation techniques must be applied, such as introducing dummy variables or 2sls.
Since the "best" estimators are designed to use these assumptions, if some of the assumptions are violated, the "best" estimators "get hurt" more seriously than other estimators. Thus two kinds of special estimators should be considered and we call them "robust" estimators. One kind of these estimators is an estimator that is not sensitive to the violations of the assumptions. We could also think about a very common situation where we are not able to know if any of the assumptions are violated because of limited information or sample size. At this point, we may prefer a less "best" estimator that are neither as good as the "best" ones, nor sensitive to the violations of assumptions, i.e., some violations of the assumptions that a estimator may have to suffer does not screw up all the models. Or it can be considered as "least worst" estimator. The first kind of robust, i.e. insensitive estimator, is very common, for an example,the OLS estimator itself is considered as a robust estimator.
Another kind of robust estimator is designed to resist the violations. I believe if you have taken a intermediate level econometric training, you must know such estimators, but you may not know it is called "robust estimator". The most commonly seen "robust" estimator is the ones used to correct heteroscedasticity. One example is about the structural changes. We cannot estimated the var-cov matrix directly. We have to estimated two var-cov matrices and using some kind of weigh techniques to jointly determine the estimate of var-cov matrix. This example was seen in an exam of my masters econometric class.
To be continued...
