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 assumptions of departure.
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 formula that a theory can be fulfilled. For example: “Assuming Messi is not injured, he will play his 100 match with Barcelona.”
We have a starting hypothesis and a result. The hypothesis is not to be injured. If he is injured, the prediction that he will play his 100th match in the league will not be fulfilled. In this case, we are not working with a robust estimator. Why? Because if it were a robust estimator the fact that he had an injury would not jeopardize the prediction.
The robust estimator and the assumptions of departure
The previous example is a frankly simple example. In statistics, unless we have basic knowledge, they are not such easy examples. However, we will try to explain the starting assumption that is usually broken when we make an estimate.
The initial assumptions or initial assumptions are common in economics. It is very common for an economic model to specify initial assumptions. For example, supposing that a market is perfectly competitive is common in many economic models.
In the case of assuming that we are facing a perfectly competitive market, we are assuming – simplifying a lot – that we are all the same. We all have the same money, the products are the same and nobody can influence the price of a good or service.
Under this prism, in statistics, the starting assumption that stands out above all others is the probability distribution. For certain properties of our estimator to be met, it must be fulfilled that the phenomenon to be studied is distributed according to a structure of probabilities.
The normal type probability distribution is the most common. Hence its name. It is called that because it is “normal” or usual. It is very frequent to see how in many statistical studies it is indicated: “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 give an example. Imagine that we want to calculate the average goals per season of Leo Messi. In our study, we assume that the probability distribution of Messi’s goals is a normal distribution. So, we use an average estimator. That estimator has a formula. We apply it and it gives us a result. For example, 48.5 goals per season.
Given the above, suppose we were wrong in the type of probability distribution. If the probability distribution were actually a student t-distribution, applying the corresponding mean formula would it 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 mistakes in the initial assumption does not significantly alter the results.