  Statistical depth, a commonly used analytic tool in non-parametric
statistics, has been extensively studied for multivariate and functional
observations over the past few decades. Although various forms of depth were
introduced, they are mainly procedure-based whose definitions are independent
of the generative model for observations. To address this problem, we introduce
a generative model-based approach to define statistical depth for both
multivariate and functional data. The proposed model-based depth framework
permits simple computation via Monte Carlo sampling and improves the depth
estimation accuracy. When applied to functional data, the proposed depth can
capture important features such as continuity, smoothness, or phase
variability, depending on the defining criteria. Specifically, we view
functional data as realizations from a second-order stochastic process, and
define their depths through the eigensystem of the covariance operator. These
new definitions are given through a proper metric related to the reproducing
kernel Hilbert space of the covariance operator. We propose efficient
algorithms to compute the proposed depths and establish estimation consistency.
Through simulations and real data, we demonstrate that the proposed functional
depths reveal important statistical information such as those captured by the
median and quantiles, and detect outliers.
