L-estimator

In statistics, an L-estimator is an estimator which is an L-statistic – a linear combination of order statistics of the measurements. 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 ${\displaystyle x_{1},\ldots ,x_{n}}$, if ${\displaystyle n=2k+1}$ is odd, the median equals ${\displaystyle x_{(k+1)}}$, the ${\displaystyle (n+1)/2}$-th order statistic; if ${\displaystyle n=2k}$ is even, it is the average of two order statistics: ${\displaystyle (x_{(k)}+x_{(k+1)})/2}$. These are both linear combinations of order statistics, and the median is therefore a simple example of an L-estimator.

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