# Breaking point

The break point (engl point break down.) Is a measure of the robustness of an estimator . The breakpoint indicates the portion of the data that is required to falsify the result of the estimator as desired. One then says that the estimator breaks down.

## Finite breaking point

The finite break point is the smallest fraction of observations that causes the estimator to collapse. He is designated with . ${\ displaystyle n}$ ${\ displaystyle \ varepsilon _ {n} ^ {*}}$ The arithmetic mean z. B. has a finite breaking point of , since a sufficiently large outlier is enough to increase its value at will. With the -trimmed mean, more than samples must already be outliers in order to have an influence on the estimate. So the breaking point is included ${\ displaystyle \ varepsilon _ {n} ^ {*} = {\ frac {1} {n}}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ lfloor \ alpha n \ rfloor}$ ${\ displaystyle \ varepsilon _ {n} ^ {*} = {\ frac {\ lfloor \ alpha n \ rfloor +1} {n}}}$ ## Asymptotic break point

The asymptotic break point, usually simply referred to as the break point, indicates the relative proportion of the data required to falsify the estimate. It is usually referred to with . It is obtained by letting the number of observations at the finite breaking point tend towards infinity, i.e. ${\ displaystyle \ varepsilon ^ {*}}$ ${\ displaystyle \ varepsilon ^ {*} = \ lim _ {n \ to \ infty} \ varepsilon _ {n} ^ {*}}$ This results in a breaking point of for the arithmetic mean and a breaking point of for the trimmed mean${\ displaystyle \ varepsilon ^ {*} = 0}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ varepsilon ^ {*} = \ alpha}$ 