A robust estimator or one that has the property of robustness, is one whose validity is not altered as a result of the violation of any of the starting assumptions.
The idea of a robust estimator is to prepare for possible failures in the initial assumptions. In statistics and economics, initial hypotheses are normally used. That is, assumptions under which a formulates that a theory can be fulfilled. For example: "Assuming Messi is not injured, he will play his 100th game with Barcelona."
We have a starting hypothesis and a result. The hypothesis is that he does not injure himself. If he is injured, the prediction that he will play his 100th league game will not come true. In this case, we are not working with a robust estimator. Why? Because if he were a robust estimator, the fact that he had an injury would not jeopardize the prediction.
The robust estimator and the starting assumptions
The example above is a frankly simple example. In statistics, unless we have basic knowledge, they are not such easy examples. However, we are going to try to explain the initial assumption that is usually broken when we make an estimate.
The starting assumptions or initial assumptions are common in economics. It is very common for an economic model to specify initial assumptions. For example, assuming that a market is perfectly competitive is common in many economic models.
In the case of supposing that we are facing a perfectly competitive market, we are supposing - simplifying a lot - that we are all the same. We all have the same money, the products are the same, and no one can influence the price of a good or service.
From this perspective, in statistics, the starting assumption that stands out above all others is that of the probability distribution. For certain properties of our estimator to be fulfilled, it must be fulfilled that the phenomenon to be studied is distributed according to a probability structure.
The normal probability distribution is the most common. Hence its name. It is so called because it is "normal" or usual. It is very frequent, to see how in many statistical studies it is stated: "We assume that the random variable X is normally distributed."
Under the normal distribution, there are some estimators that work well. Of course, we must ask ourselves what if the distribution of the random variable X is not a normal distribution? It could be for example a hypergeometric distribution.
Robust estimator example
Now that we have a slight idea, let's take an example. Let's imagine that we want to calculate the average of Leo Messi's goals per season. In our study, we assume that the probability distribution of Messi's goals is a normal distribution. So we use an estimator of the mean. That estimator has a formula. We apply it and it gives us a result. For example, 48.5 goals per season.
Taking into account the above, suppose that we have made a mistake in the type of probability distribution. If the probability distribution were actually a student's t-distribution, would applying the corresponding mean formula give us the same result? For example, the result may be 48 goals. The result is not the same, however, we have come very close. In conclusion, we could say that the estimator is robust since making a mistake in the initial assumption does not significantly alter the results.