I am looking for a robust version of Hotelling's $T^2$ test for the mean of a vector. As data, I have a $m\ \times\ n$ matrix, $X$, each row an i.i.d. sample of an $n$-dimensional RV, $x$. The null hypothesis I wish to test is $E[x] = \mu$, where $\mu$ is a fixed $n$-dimensional vector. The classical Hotelling test appears to be susceptible to non-normality in the distribution of $x$ (just as the 1-d analogue, the Student t-test is susceptible to skew and kurtosis).

what is the state of the art robust version of this test? I am looking for something relatively fast and conceptually simple. There was a paper in COMPSTAT 2008 on the topic, but I do not have access to the proceedings. Any help?

  • $\begingroup$ thanks for editing to have proper math. I couldn't find info on how to render math in 'markdown' language. looks just like raw \LaTeX does it? $\endgroup$ – shabbychef Aug 6 '10 at 21:33
  • $\begingroup$ Yes, it is Latex. See this meta thread: meta.stats.stackexchange.com/questions/218/… $\endgroup$ – user28 Aug 6 '10 at 21:46

Sure: two answers

a) If by robustness, you mean robust to outliers, then run Hottelling's T-test using a robust estimation of scale/scatter: you will find all the explications and R code here: http://www.statsravingmad.com/blog/statistics/a-robust-hotelling-test/

b) if by robustness you mean optimal under large group of distributions, then you should go for a sign based T2 (ask if this what you want, by the tone of your question i think not).

PS: this is the paper you want; Roelant, E., Van Aelst, S., and Willems, G. (2008), “Fast Bootstrap for Robust Hotelling Tests,” COMPSTAT 2008: Proceedings in Computational Statistics (P. Brito, Ed.) Heidelberg: Physika-Verlag, to appear.

  • $\begingroup$ yes, that is the paper I am looking for, but I cannot find a version of it online. The R-code might be useful, but I am working in Matlab, and don't have time to do the translation (the R code looks nontrivial). I would even be willing to settle for the bibliography of this paper at the moment. $\endgroup$ – shabbychef Aug 10 '10 at 0:18

Some robust alernatives are discussed in A class of robust stepwise alternativese to Hotelling's T 2 tests, which deals with trimmed means of the marginals of residuals produced by stepwise regression, and in A comparison of robust alternatives to Hoteslling's T^2 control chart, which outlines some robust alternatives based on MVE, MCD, RMCD and trimmed means.


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