Non-Euclidean data are now prevalent in many fields, necessitating the development of new concepts such as distribution functions, quantiles, rankings, and signs of these data in order to make nonparametric statistical inferences. I’m here. This study provides new ideas about local and global quantiles in metric space. This is achieved by extending the 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 central outward ordering of the metric space induced by the local and global quantiles. Theoretical properties such as root-$n$ consistency and uniform consistency of local and global empirical quantiles, and rank and sign distribution degrees of freedom are established. The median empirical metric, defined here as the 0th empirical global metric quantile, has been proven to be resistant to contamination by both theoretical and numerical approaches. Quantiles have been shown to be of value through extensive simulations in many metric spaces. Furthermore, we introduce a family of fast rank-based independence tests for common metric spaces. Monte Carlo experiments show excellent finite-sample performance of the test. Quantile values are presented in a real-world setting by analyzing hippocampal data.
