Definition
There are various definitions of a "robust statistic". Strictly speaking, a robust statistic is resistant to errors in the results, produced by deviations from assumptions (e.g. of normality). This means that if the assumptions are only approximately met, the robust estimator will still have a reasonable efficiency, and reasonably small bias, as well as being asymptotically unbiased, meaning having a bias tending towards 0 as the sample size tends towards infinity.
One of the most important cases is distributional robustness. Classical statistical procedures are typically sensitive to "longtailedness" (e.g., when the distribution of the data has longer tails than the assumed normal distribution). Thus, in the context of robust statistics, distributionally robust and outlier-resistant are effectively synonymous. For one perspective on research in robust statistics up to 2000, see Portnoy and He (2000).
A related topic is that of resistant statistics, which are resistant to the effect of extreme scores.
Read more about this topic: Robust Statistics
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