On a Notion of Data Depth Based on Random Simplices

Liu, Regina Y.

Ann. Statist., Tome 18 (1990) no. 1, p. 405-414 / Harvested from Project Euclid

Résumé

For a distribution $F$ on $\mathbb{R}^p$ and a point $x$ in $\mathbb{R}^p$, the simplical depth $D(x)$ is introduced, which is the probability that the point $x$ is contained inside a random simplex whose vertices are $p + 1$ independent observations from $F$. Mathematically and heuristically it is argued that $D(x)$ indeed can be viewed as a measure of depth of the point $x$ with respect to $F$. An empirical version of $D(\cdot)$ gives rise to a natural ordering of the data points from the center outward. The ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and $L$-statistics. This generalized sample median and $L$-statistics are affine equivariant.

Détail

Publié le : 1990-03-14
Classification:  Simplex,  simplicial depth,  multivariate median,  $L$-statistics,  angularly symmetric distributions,  location estimators,  consistency,  affine equivariance,  62H05,  62H12,  60D05