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 x% trimmed version is obtained by discarding the x% lowest or highest observations or on both end: 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 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 Interdecile range, the 10% trimmed range Trimmed estimators involving only linear combinations of points are examples of L-estimators. Most often, trimmed estimators are used for parameter estimation of the same parameter as the untrimmed estimator.

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Midhinge

In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator. The midhinge is related to the interquartile range (IQR), the difference of the third and first quartiles (i.e. ), which is a measure of statistical dispersion. The two are complementary in sense that if one knows the midhinge and the IQR, one can find the first and third quartiles.

L-estimator

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.

Robust measures of scale

In statistics, robust measures of scale are methods that quantify the statistical dispersion in a sample of numerical data while resisting outliers. The most common such robust statistics are the interquartile range (IQR) and the median absolute deviation (MAD). These are contrasted with conventional or non-robust measures of scale, such as sample standard deviation, which are greatly influenced by outliers.

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