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In [[statistics]], a '''trimmed estimator''' is an [[estimator]] derived from another estimator by excluding some of the [[extreme value]]s, a process called [[Truncation (statistics)|truncation]]. This is generally done to obtain a more [[robust statistic]], and the extreme values are considered [[outlier]]s. Trimmed estimators also often have higher [[Efficiency (statistics)|efficiency]] for [[mixture distribution]]s and [[heavy-tailed distribution]]s than the corresponding untrimmed estimator, at the cost of lower efficiency for other distributions, such as the [[normal distribution]].
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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 2.5% to 97.5% range. In some cases a trimmed estimator discards a fixed number of points (such as maximum and minimum) instead of a percentage.
 
==Examples==
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 mean]]
* [[Modified mean]], discarding the minimum and maximum values
* [[Interquartile mean]], the 25% [[trimmed mean]]
* [[Midhinge]], the 25% trimmed [[mid-range]]
Trimmed estimators used to estimate a [[scale parameter]] include:
* [[Interquartile range]], the 25% trimmed [[range (statistics)|range]]
* [[Interdecile range]], the 10% trimmed range
 
Trimmed estimators which only involve linear combinations of points are examples of [[L-estimator]]s.
 
==Applications==
===Estimation===
Most often, trimmed estimators are used for [[parameter estimation]] of the same parameter as the untrimmed estimator. In some cases the estimator can be used directly, while in other cases it must be adjusted to yield an [[unbiased estimator|unbiased]] [[consistent estimator]].
 
For example, when estimating a [[location parameter]], for a symmetric distribution a trimmed estimator will be unbiased (assuming the original estimator was unbiased), as it removes the same amount above and below. However, if the distribution has [[skewness|skew]], trimmed estimators will generally be biased and require adjustment. For example, in a skewed distribution, the [[nonparametric skew]] (and [[Pearson's skewness coefficients]]) measure the bias of the median as an estimator of the mean.
 
When estimating a [[scale parameter]], using a trimmed estimator as a [[robust measures of scale]], such as to estimate the [[population variance]] or population [[standard deviation]], one generally must multiply by a [[scale factor]] to make it an unbiased consistent estimator; see [[scale parameter#Estimation|scale parameter: estimation]].
 
For example, dividing the IQR by <math>2\sqrt{2} \operatorname{erf}^{-1}(1/2) \approx 1.349</math> (using the [[error function]]) makes it an unbiased, consistent estimator for the population standard deviation if the data follow a [[normal distribution]].
 
===Other uses===
Trimmed estimators can also be used as statistics in their own right – for example, the median is a measure of location, and the IQR is a measure of dispersion. In these cases, the sample statistics can act as estimators of their own [[expected value]]. For example, the MAD of a sample from a standard [[Cauchy distribution]] is an estimator of the population MAD, which in this case is 1, whereas the population variance does not exist.
 
==See also==
* [[Winsorising]], a related technique
* [[Core inflation]], an economic statistic that omits volatile components
 
==References==
{{unreferenced|date=April 2013}}
 
[[Category:Estimation theory]]
[[Category:Robust statistics]]

Revision as of 09:49, 21 February 2014

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