Defining extremes and trimming by minimum covering sets
Given a class \(\{\) \(R\}\) of convex subsets of \(R^ d\), the author defines extremes in a d-variate sample as the points of the sample that lie on the boundary of the smallest set R which contains the whole sample. Corresponding trimmed sums are then sums of all sample points omitting layers of the extremes. Representations for the distributions of these trimmed sums are obtained and used to establish necessary and sufficient conditions for consistency and asymptotic normality if the number of points removed remains bounded in probability as the sample size increases. It turns out that, as in the one-dimensional case, this form of light trimming does not improve the asymptotic behavior of the sample mean. The general approach presented in the paper contains as special cases the minimum covering ellipse of a sample and trimming by polyhedra.
scientific article; zbMATH DE number 4153570
scientific article