# L-estimator

**L-estimation**

In statistics, an L-estimator is an estimator which is an L-statistic – a linear combination of order statistics of the measurements.wikipedia

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### Mid-range

**midsummarymidrangehalf-range**

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.

However, it finds some use in special cases: it is the maximally efficient estimator for the center of a uniform distribution, trimmed mid-ranges address robustness, and as an L-estimator, it is simple to understand and compute.

### M-estimator

**M-estimationestimation**

However, they are inefficient, and in modern times robust statistics M-estimators are preferred, though these are much more difficult computationally.

are called M-estimators ("M" for "maximum likelihood-type" (Huber, 1981, page 43)); other types of robust estimator include L-estimators, R-estimators and S-estimators.

### Trimmed estimator

**trimmedtrimming**

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.

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

### Midhinge

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.

Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.

### Order statistic

**order statisticsorderedth-smallest of items**

In statistics, an L-estimator is an estimator which is an L-statistic – a linear combination of order statistics of the measurements.

L-estimator – linear combinations of order statistics

### Efficiency (statistics)

**efficientefficiencyinefficient**

However, they are inefficient, and in modern times robust statistics M-estimators are preferred, though these are much more difficult computationally.

A more traditional alternative are L-estimators, which are very simple statistics that are easy to compute and interpret, in many cases robust, and often sufficiently efficient for initial estimates.

### Robust statistics

**robustbreakdown pointrobustness**

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.

L-estimators are a general class of simple statistics, often robust, while M-estimators are a general class of robust statistics, and are now the preferred solution, though they can be quite involved to calculate.

### Trimean

Like the median and the midhinge, but unlike the sample mean, it is a statistically resistant L-estimator with a breakdown point of 25%.

### Range (statistics)

**rangerangingsample range**

In particular, the range is a linear function of order statistics, which brings it into the scope of L-estimation.

### Box plot

**boxplotbox and whisker plotadjusted boxplots**

However, the simplicity of L-estimators means that they are easily interpreted and visualized, and makes them suited for descriptive statistics and statistics education; many can even be computed mentally from a five-number summary or seven-number summary, or visualized from a box plot.

In addition to the points themselves, they allow one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean.

### Robust measures of scale

**Qn estimatorrobust estimator of dispersionrobust measure of scale**

Robust L-estimators used to measure dispersion, such as the IQR, provide robust measures of scale.

One of the most common robust measures of scale is the interquartile range (IQR), the difference between the 75th percentile and the 25th percentile of a sample; this is the 25% trimmed range, an example of an L-estimator.

### Five-number summary

**five-number summaries**

However, the simplicity of L-estimators means that they are easily interpreted and visualized, and makes them suited for descriptive statistics and statistics education; many can even be computed mentally from a five-number summary or seven-number summary, or visualized from a box plot.

In addition to the points themselves, many L-estimators can be computed from the five-number summary, including interquartile range, midhinge, range, mid-range, and trimean.

### L-moment

**probability weighted moments**

Sample L-moments are L-estimators for the population L-moment, and have rather complex expressions.

*L-estimator

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, an L-estimator is an estimator which is an L-statistic – a linear combination of order statistics of the measurements.

### Estimator

**estimatorsestimateestimates**

In statistics, an L-estimator is an estimator which is an L-statistic – a linear combination of order statistics of the measurements.

### Descriptive statistics

**descriptivedescriptive statisticstatistics**

However, the simplicity of L-estimators means that they are easily interpreted and visualized, and makes them suited for descriptive statistics and statistics education; many can even be computed mentally from a five-number summary or seven-number summary, or visualized from a box plot. They thus are useful in robust statistics, as descriptive statistics, in statistics education, and when computation is difficult.

### Statistics education

**statistics educatorstatisticsPh.D. Statistics**

However, the simplicity of L-estimators means that they are easily interpreted and visualized, and makes them suited for descriptive statistics and statistics education; many can even be computed mentally from a five-number summary or seven-number summary, or visualized from a box plot. They thus are useful in robust statistics, as descriptive statistics, in statistics education, and when computation is difficult.

### Median

**averagesample medianmedian-unbiased estimator**

A basic example is the median.

### Quantile

**quantilesquintiletertile**

### Interquartile range

**inter-quartile rangebelowinterquartile**

### Interdecile range

### Truncated mean

**trimmed meanmodified mean**

### Interquartile mean

**interquartile**

### Winsorized mean

### Central tendency

**Localitycentral locationcentral point**

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.