An estimator is a statistician who is required certain conditions so that he can calculate certain parameters of a population with certain guarantees.
That is, an estimator is a statistician . Now, he is not just any statistician. It is a statistician with certain properties. An example could be the mean or variance . These well-known metrics are estimators.
We name these two as being the simplest, but in statistics there are many more. Now, going back to the definition, what do we understand by certain conditions so that certain parameters can be calculated with certain guarantees?
First, we must understand that when we conduct a research study, we usually want to study a certain parameter. For example, we want to study the average height of trees in a certain city in Colombia. The variable under study is the height of the trees in a certain city in Colombia. While, the parameter is the average height of the trees in that city.
In the previous example, what condition would we have to demand from our estimator? Well, for example, do not take negative values. And, of course, that the calculation of the average height of place at possible values. If the tallest tree measures 10 meters, the average estimator cannot throw us 15 meters. In that case, it could not be an estimator, since it would not be giving rise to physically possible values.
Thus, from the above we conclude that the estimators are statistics that must, necessarily, take possible values from the data we are studying.
Now, it is not enough just to take values that are within the data range. Normally certain properties are required so that we have certain guarantees. It may be the case that certain estimators meet the condition of being estimators, but if they estimate poorly, they will be qualified as bad estimators.
Recommended properties of an estimator
In order to fulfill its function well, in addition to estimators meeting their basic condition of estimators, it is recommended that they meet certain additional properties. These properties are what will allow the conclusions drawn from our study to be reliable.
- Sufficient:The sufficiency property indicates that the estimator works with all the sample data. For example, the average does not choose only 50% of the data. It takes into account 100% of the data to calculate the parameter.
- Insesgado:The property of insecurity refers to the centrality of an estimator. That is, the average of an estimator must match the parameter to be estimated. We should not confuse the average of an estimator with the average estimator.
- Consistent:The concept of consistency goes hand in hand with the sample size and the limit concept. In simple words, he comes to tell us that the estimators fulfill this property when, in case the sample is very large, they can estimate almost without error.
- Efficient:The efficiency property can be absolute or relative. An estimator is efficient in an absolute sense when the variance of the estimator is minimal. We must not confuse variance of an estimator with variance estimator.
- Robust:It is said that an estimator is robust in case, despite the fact that the starting hypothesis is incorrect, the results closely resemble the real ones.
The above properties are the main ones. Of course, within each property there are many different cases. Similarly, there are also other desirable properties.
Other desirable properties of estimators
An example of desirable property is that of invariant with changes of scale. This property indicates that, in case of changing the unit of measure, the value to be estimated does not change. For example, if we measure the trees in centimeters and then in meters, the average value should be the same. With which, we could say that the mean is an invariant estimator before changes of scale.
Another property that statistics manuals usually indicate is that of invariant before changes of origin. To continue with the previous case, we will see a hypothetical case. Suppose that after measuring all the trees, we conclude that we must add 10 centimeters to the recorded height of each tree. The ribbon used was poorly measured and we have to make this change to adjust the data to reality. What we are doing is a change of origin. And the question is, will the average height result change?
Unlike in the change of scale, here the change of origin does affect. If it turns out that all trees measure 10 centimeters more, then the average height will rise.
Therefore, we can say that the mean is an invariant estimator before changes of scale but variant before changes of origin.