Plug‐in Estimation of General Level Sets

Summary Given an unknown function (e.g. a probability density, a regression function, …) f and a constant c, the problem of estimating the level set L(c) ={f≥c} is considered. This problem is tackled in a very general framework, which allows f to be defined on a metric space different from . Such a degree of generality is motivated by practical considerations and, in fact, an example with astronomical data is analyzed where the domain of f is the unit sphere. A plug‐in approach is followed; that is, L(c) is estimated by Ln(c) ={fn≥c}, where fn is an estimator of f. Two results are obtained concerning consistency and convergence rates, with respect to the Hausdorff metric, of the boundaries ∂Ln(c) towards ∂L(c). Also, the consistency of Ln(c) to L(c) is shown, under mild conditions, with respect to the L1 distance. Special attention is paid to the particular case of spherical data.