Plug‐in Estimation of General Level Sets

Abstract

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 R d . 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 L n (c) = {f n ≥ c}, where f n is an estimator of f . Two results are obtained concerning consistency and convergence rates, with respect to the Hausdorff metric, of the boundaries ∂L n (c) towards ∂L(c). Also, the consistency of L n (c) to L(c) is shown, under mild conditions, with respect to the L 1 distance. Special attention is paid to the particular case of spherical data.

Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

“…An alternative approach, based on the geometric properties of the compact support sets, has been presented by Hartigan [20], Cuevas and Fraiman [11], Cuevas and Rodríguez-Casal [13]. The problem of estimating general level sets under compactness assumptions has been discussed by Cuevas et al [12]. The asymptotic behavior of minimum volume sets and of a generalized quantile process is analyzed by Polonik [26].…”

“…see Baíllo et al [3]; Rigollet and Vert [27]; Cuevas et al [12]), that is, L(c) is estimated by L n (c) = {F n (x) ≥ c}, for c ∈ (0, 1), where F n is a consistent estimator of F . The regularity properties of F and F n as well as the consistency properties of F n will be specified in the statements of our theorems.…”

Estimating covariate functions associated to multivariate risks: a level set approach

“…Note that, as in Theorem 3 in Cuevas et al [9], Theorem 3.1 above does not require any continuity assumption on F n . Furthermore, we remark that a sequence T n whose divergence rate is large, implies a convergence rate p n quite slow.…”

“…Rigollet and Vert [26]; Cuevas et al [9]), we need to work in a non-compact setting and this requires special attention in the statement of our problem.…”

Estimation of density level sets with a given probability content

“…Under some assumptions, consistency and rates of convergence (for the volume of the symmetric difference) have been established in Baillo, Cuevas, and Justel (2000), Baillo, Cuestas-Alberto, and Cuevas (2001), and Baillo (2003), and an exact convergence rate is obtained in Cadre (2006). Recently, Mason and Polonik (2009) derive the asymptotic normality of the volume of the symmetric difference for kernel plug-in level set estimates (see also related works in Molchanov 1998;Cuevas, Gonzalez-Manteiga, and Rodriguez-Casal 2006). So far, most theoretical works on the subject have focused on the estimation of a density level set at a fixed threshold t in Equation (1).…”