An estimator is a statistic that requires certain conditions to be able to calculate certain parameters of a population with certain guarantees.
That is, an estimator is a statistic. Now, he is not just any statistician. It is a statistic with certain properties. An example could be the mean or the variance. These well-known metrics are estimators.
We name these two because they are 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 of all, we must understand that when we conduct a research study, we normally want to study a certain parameter. For example, we want to study what is 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. Whereas, the parameter is the average height of the trees in that city.
In the example above, what condition would we have to require from our estimator? Well, for example, do not take negative values. And, of course, that the calculation of the average height leads to possible values. If the tallest tree is 10 meters, the mean estimator cannot give 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 statisticians 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 of you in order for us to have certain guarantees. It may be the case that certain estimators meet the condition of being estimators, but if they estimate badly, they will be classified as bad estimators.
Recommended properties of an estimator
In order for it to fulfill its function well, in addition to the estimators fulfilling their basic condition of estimators, it is recommended that they fulfill 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 data in the sample. For example, the mean does not pick only 50% of the data. It takes into account 100% of the data to calculate the parameter.
- Unbiased: The unbiased property refers to the centrality of an estimator. That is, the mean of an estimator must coincide with the parameter to be estimated. We should not confuse the mean of an estimator with the mean estimator.
- Consistent: The concept of consistency is coupled with the size of the sample and the concept of limit. In simple words, it comes to tell us that the estimators fulfill this property when, in the case of a very large sample, they can estimate almost without error.
- Efficient: The efficiency property can be absolute or relative. An estimator is efficient in the absolute sense when the variance of the estimator is minimal. We must not confuse variance of an estimator with a variance estimator.
- Robust: An estimator is said to be robust in the event that, despite the initial hypothesis being 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. Likewise, there are other desirable properties as well.
Other desirable properties of estimators
An example of a desirable property is that of invariant to changes in scale. This property indicates that, if the unit of measurement is changed, the value to be estimated does not change. For example, if we measure trees in centimeters and then in meters, the mean value should be the same. With which, we could say that the mean is an invariant estimator before scale changes.
Another property that statistics manuals usually indicate is that of invariant to changes in origin. To continue with the previous case, we are going to 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 strip 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 result of mean height change?
Contrary to the change of scale, here the change of origin does affect. If it turns out that all the trees are 10 centimeters taller, 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.