Asymptotics of stationary points of local simplicial depth in the univariate case

Abstract

Local depth (see Agostinelli and Romanazzi, 2011) extends the usual depth definition so as to account for multimodal distributions and clustered data. Applications include rankings of data according to centrality in suitable neighborhoods and locations of partial centers. Here the basic definitions and properties of local simplicial depth in the univariate case are reviewed. The univariate case is special because the derivative of the depth function has a simplemathematical representation. Using this result and the theory of empirical processes, the asymptotic behavior of the stationary points of local simplicial depth is further investigated with some new findings. Illustrations based on simulated and real data are given.