Depth functions for tree-indexed data

G. Francisci${}^a$ and A. N. Vidyashankar${}^b$

${}^a$University of Trento, ${}^b$George Mason University

Depth functions quantify the degree of centrality of a point relative to a multivariate data set, with higher depth indicating greater centrality and lower depth indicating peripheral or outlying positions. They are important tools in non-parametric and robust statistics and have been used for classification and clustering. Several extensions exist for functional and metric space valued data. In this work, we introduce depth functions for tree-indexed data. Our analysis is based on the intensity measure of the point processes generating the data. When the point processes have independent and identically distributed components, the depth function reduces to the depth of a single component. We investigate the statistical properties and asymptotic behavior of the depth function. This enables the study of medians and quantiles of tree-indexed data. Finally, we apply these results to the classification of tree-indexed data using an analog of the depth-versus-depth (DD) classifier.

Keywords: Classification, Depth functions, Tree-indexed data