Nonparametric Functional Quantiles, Skewness, and Probability Bands via the Functional Signed Directionality
We propose the Functional Signed Directionality (FuSD), a nonparametric order for function-valued random elements that defines a pushforward distribution on the real line. This construction enables probabilistic inference in the functional domain: we define set-valued functional quantiles, introduce quantile-based summaries of functional spread and skewness, and construct tight probability bands that reflect the underlying functional law via the FuSD distribution.
Overview
For functional bands representing central regions, we formalize a coverage condition and adapt an Fβ score to jointly assess coverage and informativeness. Our framework is built under minimal assumptions, designed to account for the geometry of the ambient space and to reflect the distribution of the underlying random element. The approach is simultaneous rather than pointwise, avoids heavy pre-smoothing and outlier trimming, which makes it suitable for irregular and noisy sampling. We establish fundamental properties that allow FuSD to represent the functional distribution, and study the asymptotic behavior of FuSD-induced functional quantiles through the almost-sure inclusion of the limsup of empirical quantiles. An environmental application demonstrates that FuSD delivers interpretable quantiles, calibrated tight bands, and meaningful functional spread and skewness summaries.
Presenters
Brief Biography
Dr. Emmanuel Ambriz is a Ph.D. in Statistics whose research has focused on frontier challenges in copula theory, particularly in multivariate vine copula models, and their relevance to other branches of modern statistics.
Dr. Ambriz has recently joined the CEMSE Division as a postdoctoral fellow. He obtained his M.S. and Ph.D. degrees from the Centro de Investigación en Matemáticas (CIMAT), Mexico, in 2016 and 2024, respectively. From 2017 to 2022, he has worked as a Professor and Researcher at the Universidad Regional Amazónica Ikiam in Ecuador, where he has been involved in several research projects related to conservation and water resource challenges in the Ecuadorian Amazon.
In addition to his research career, Dr. Ambriz has actively collaborated as a statistical consultant with various industries, public institutions, and NGOs in Mexico and Ecuador, applying statistical methods to support data-driven decision-making across diverse sectors.
Related People
Related Researchers
Emmanuel Ambriz
- Postdoctoral Research Fellow, Statistics