In contrast to the many threads on this site that recommend Dixon's and Grubb's tests, the author of one answer, at this thread, contends that "Really, these have been discredited long ago" and advocates 2 other methods. I don't feel qualified to sort out these arguments, but I would like to ask whether there is any consensus among statisticians as to the merits of either position.
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The issue becomes less contentious if you state the facts. After all, all multivariate robust estimation procedures have at their core an outlier detection algorithm and all will in some form or another output a list of suspect observations. Stated otherwise, given a robust fit, identifying outliers is in principle not an issue.
The main difference between robust estimation approaches and the testing approaches (Dixon, Grubbs) is that the latter can sustain at most a single outlier. In contrast, most state of the art robust estimation procedures have been designed to handle nearly 50% contamination (they can in principle be tuned to handle anywhere between 0 and nearly 50 percent outliers trading off robustness for computational costs).
 Rousseeuw P. J. and Van Zomeren B. C., Unmasking Multivariate Outliers and Leverage Points.
I say no! I have done research on Dixon's test back in the 1980s. I took a look at that post and commented there. I think there is confusion because robust estimation and outlier detection though similar have different objectives and some people I think seem to feel that since the outlier methods are not mentioned in the robustness literature that there is something wrong with them. I hope other will agree with me in answering this question.