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Halfspace depth: The geometry of multivariate quantiles

Stanislav Nagy, Charles University


Statistical depth is a non-parametric tool applicable to multivariate data, whose main goal is a reasonable generalisation of quantiles to multivariate datasets. We discuss the halfspace depth, the most important depth in statistics. This depth was first proposed in 1975; its rigorous investigation starts in the 1990s, and still an abundance of open problems stimulates the research in the area. We present several interesting links of the halfspace depth and well-studied concepts from geometry. Using these relations we resolve several open problems concerning the depth, and outline perspectives for future research both in the area of depth and in convex geometry. The talk is intended to be largely self-contained; no particular knowledge of probability, statistics, or geometry is necessary.