kvantilien

Kvantilien is a concept in statistics used to describe generalized multivariate quantiles. It does not denote a single universally accepted object, but rather a family of constructions that aim to summarize the distribution of a multivariate variable by a set or curve of representative points corresponding to specified probability levels. The idea extends the one-dimensional notion of a quantile to higher dimensions, where there is no natural linear order.

In higher dimensions, kvantilien constructions address the non-uniqueness of multivariate quantiles by relying on different reference frameworks. Common approaches include depth-based methods that define contours enclosing central mass, projection quantiles that analyze quantiles along chosen directions, and spatial or ball-quantile concepts that associate probability mass with particular points or regions. Each definition yields different geometric shapes and transformation properties, but all share the goal of providing a structured view of central tendency and tails in a multivariate distribution.

Properties and computation vary across definitions. Some kvantilien constructions are affine-invariant, meaning they behave predictably under linear transformations, while others prioritize simplicity or interpretability. For a fixed probability level p, the kvantilien may appear as a curve, a closed region, or a finite set of points in the feature space. Empirical estimation from data typically involves solving a sequence of one-dimensional problems along projections or solving optimization problems that minimize a chosen quantile loss.

Applications include multivariate data visualization, anomaly detection, and risk assessment in finance and environmental studies. See also quantile, multivariate quantiles, data depth, spatial quantiles, and projection quantiles.