# Optimizing Robust Statistics

A robust paired t-test is a better choice for skewed distributions than the conventional paired t-test (e.g Fradrette, Keselman, Lix, & Wilcox, 2003). One version of the robust test uses a trimmed mean and a Winsorized variance. Is there a way to find the optimal cutoff point for the Winsorization percentiles that maximises the sensitivity?

If I understand correctly, robust statistics perform better for skewed distributions than for symmetric ones and modifying the tails of the distributions using Winsorization or trimming thus changes the skewness. For example, consider if one were to increase the Winsorization percentage iteratively. I imagine that the skewness statistic approaches a certain criterion and the adjusted $R^2$ difference for the cross-correlation between the original distribution and the current Winsorized distribution approaches a criterion close to zero.

If using such an algorithm, would it be possible to reach an unbiased estimates of central tendency and measures of dispersion? Are there any known implementations, for example, in the field of machine learning?

Fradrette, K., Keselman, H. J., Lix, L., Algina, J., & Wilcox, R. R. (2003). Conventional and robust paired independent-samples t test: Type I error and power rate. Journal of Modern Applied Statistical Methods, 2(2) 22, 480-496. Retrieved from http://digitalcommons.wayne.edu/jmasm/vol2/iss2/22 [Open Access] 2016-08-25.

• 1. What would be optimal depends on the distribution you're sampling from (not just its skewness coefficient) ... generally the point of using a robust procedure is to choose something that does reasonably well across a variety of possibilities, not something that optimizes a particular case. (Unless I mistake your meaning and your proposing some adaptive robust procedure or something) ... 2. unbiased estimate for what? Aug 25, 2016 at 2:03
• It would be better if the authors would refer to more of the work on robust tests done outside social science journals. There's plenty of robustifications of the t-test to be found -- it would be nice to see them address how they stack up Aug 25, 2016 at 2:11
• @Glen_b I have tried to address your points. The algorithm would halt at different points for different distributions - I am not sure if this is 'adaptive', but the method should be general. Aug 25, 2016 at 2:38