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Abstract:
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Over the last two decades, various notions of data depth have emerged as powerful explanatory and inferential tools for nonparametric multivariate analysis. In this paper, we propose a new data depth, called triangle data depth, and discuss its properties. The triangle data depth is used to construct a Wilcoxon rank sum statistic for testing of equality of distribution functions in high dimensions. The performance of the test based on different notions of data depth is compared through a simulation study. Given $m$ independent $d$-dimensional observations, the sample triangle data depth enjoys computational simplicity and efficiency in high dimensions as its time complexity is $O(m^2 d)$, which is only linear in the dimension $d$, whereas many existing data depths have time complexities that are $O(m^d)$.
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