Breakdown point is a measure of the insensitivity of an estimator to multiple outliers in the data. Roughly it is given by the smallest fraction of data contamination needed to cause an arbitrarily large change in the estimate. A breakdown point of an estimator measures the ability of a statistical model to remain reliable and accurate when presented with multiple outliers in the data. In other words, it is a measure of how robust a given estimator is to outliers. 

It is commonly used as a measure of how sensitive the estimator is to extreme values or outliers in the dataset, which can often negatively impact the accuracy and reliability of estimates produced by such models. 

Explanation about Breakdown Point

The breakdown point tells us how many outlying observations we can tolerate before our estimates become seriously biased. The higher the breakdown point, the more robust an estimator will be against outliers. This means that if there are multiple outliers present in the data, an estimator with a high breakdown point will still be able to produce reliable and accurate estimates despite their presence. On the other hand, if there are multiple outliers present in the data and an estimator has low breakdown points, then its estimates could be adversely affected by these extreme values and may not be entirely reliable or accurate. Breakdown points can vary widely depending on which method or algorithm is being used for estimating parameters. Generally speaking, simple methods such as least squares regression tend to have relatively low breakdown points while more sophisticated methods such as robust regression might have relatively higher breakdown points. Additionally, some methods such as M-estimators provide higher levels of overall robustness compared to others due to their ability to adaptively down weight any deviations from what is considered “normal” behavior within a dataset. 

Advantages and Disadvantages

Breakdown point is a statistical concept that refers to the amount of contamination that a data set can withstand before the statistical results become unreliable. The advantages of using breakdown point measure in statistics are several; first, it is a measure of robustness that makes statistical methods dependable in the presence of outlying observations. Breakdown point is a key property of robust estimators that make them more efficient than traditional estimators. Secondly, the breakdown point is an indicator of how well a statistical model can handle outliers. An estimator with high breakdown point can handle a higher percentage of outliers than one with lower breakdown point.

However, the use of breakdown point methods has some disadvantages. One of the major disadvantages is that they may not be able to handle the high percentage of outliers. In statistical practice, it is often known that the majority of data may consist of outliers. If the data contains too many outliers, it may not be possible to estimate the population parameters like the mean or the median using the breakdown point method. 

Another disadvantage of breakdown point is that it may fail to detect some outliers. Breakdown point is based on a specific type of outlier, so if the data contains a different type of outlier, the breakdown point measure may not be applicable. There are different types of outliers such as global, contextual, or semi-local. Depending on the type of outliers present in the data set, the breakdown point measure may not be the best measure of robustness.

Conclusion

In conclusion, breakdown point is a measure of statistical robustness that allows for the presence of outliers in the data set. It has several advantages such as being an indicator of how well a statistical model can handle outliers. However, the use of breakdown point also has some disadvantages such as failing to detect some outliers and inability to handle high percentages of outliers. Therefore, statistical practitioners must analyze the data set to determine the type of outlier present and use the appropriate measure of robustness.