Quantiles, Ranks and Signs in Metric Spaces

Non-Euclidean data is currently prevalent in many fields, necessitating the development of novel concepts such as distribution functions, quantiles, rankings, and signs for these data in order to conduct nonparametric statistical inference. This study provides new thoughts on quantiles, both locally and globally, in metric spaces. This is realized by expanding upon metric distribution function proposed by Wang et al. (2021). Rank and sign are defined at both the local and global levels as a natural consequence of the center-outward ordering of metric spaces brought about by the local and global quantiles. The theoretical properties are established, such as the root-$n$ consistency and uniform consistency of the local and global empirical quantiles and the distribution-freeness of ranks and signs. The empirical metric median, which is defined here as the 0th empirical global metric quantile, is proven to be resistant to contaminations by means of both theoretical and numerical approaches. Quantiles have been shown valuable through extensive simulations in a number of metric spaces. Moreover, we introduce a family of fast rank-based independence tests for a generic metric space. Monte Carlo experiments show good finite-sample performance of the test. Quantiles are demonstrated in a real-world setting by analysing hippocampal data.