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In statistics, an L-estimator is an estimator which is a linear combination of order statistics of the measurements (which is also called an L-statistic). This can be as little as a single point, as in the median (of an odd number of values), or as many as all points, as in the mean. The main benefits of L-estimators are that they are often extremely simple, and often robust statistics: assuming sorted data, they are very easy to calculate and interpret, and are often resistant to outliers. They thus are useful in robust statistics, as descriptive statistics, in statistics education, and when computation is difficult. However, they are inefficient, and in modern times robust statistics M-estimators are preferred, though these are much more difficult computationally. In many circumstances L-estimators are reasonably efficient, and thus adequate for initial estimation. A basic example is the median. Given n values , if is odd, the median equals , the -th order statistic; if is even, it is the average of two order statistics: . These are both linear combinations of order statistics, and the median is therefore a simple example of an L-estimator. A more detailed list of examples includes: with a single point, the maximum, the minimum, or any single order statistic or quantile; with one or two points, the median; with two points, the mid-range, the range, the midsummary (trimmed mid-range, including the midhinge), and the trimmed range (including the interquartile range and interdecile range); with three points, the trimean; with a fixed fraction of the points, the trimmed mean (including interquartile mean) and the Winsorized mean; with all points, the mean. Note that some of these (such as median, or mid-range) are measures of central tendency, and are used as estimators for a location parameter, such as the mean of a normal distribution, while others (such as range or trimmed range) are measures of statistical dispersion, and are used as estimators of a scale parameter, such as the standard deviation of a normal distribution.

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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.

Mid-range

In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set: The mid-range is closely related to the range, a measure of statistical dispersion defined as the difference between maximum and minimum values. The two measures are complementary in sense that if one knows the mid-range and the range, one can find the sample maximum and minimum values.

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