Trimmed estimator

In statistics, a trimmed estimator is an estimator derived from another estimator by excluding some of the extreme values, a process called truncation. This is generally done to obtain a more robust statistic, and the extreme values are considered outliers. Trimmed estimators also often have higher efficiency for mixture distributions and heavy-tailed distributions than the corresponding untrimmed estimator, at the cost of lower efficiency for other distributions, such as the normal distribution.

Given an estimator, the n% trimmed version is obtained by discarding the n% lowest and highest observations: it is a statistic on the middle of the data. For instance, the 5% trimmed mean is obtained by taking the mean of the 5% to 95% range. In some cases a trimmed estimator discards a fixed number of points (such as maximum and minimum) instead of a percentage.

The median is the most trimmed statistic (nominally 50%), as it discards all but the most central data, and equals the fully trimmed mean – or indeed fully trimmed mid-range, or (for odd size data sets) the fully trimmed maximum or minimum. Likewise, no degree of trimming has any effect on the median – a trimmed median is the median – because trimming always excludes an equal number of the lowest and highest values.

Quantiles can be thought of as trimmed maxima or minima: for instance, the 5th percentile is the 5% trimmed minimum.

Trimmed estimators used to estimate a location parameter include:

Trimmed estimators used to estimate a scale parameter include:

Trimmed estimators which only involve linear combinations of points are examples of L-estimators.

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