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PLUG‐IN ESTIMATION OF GENERAL LEVEL SETS
Author(s) -
Cuevas Antonio,
GonzálezManteiga Wenceslao,
RodríguezCasal Alberto
Publication year - 2006
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/j.1467-842x.2006.00421.x
Subject(s) - mathematics , consistency (knowledge bases) , estimator , hausdorff distance , degree (music) , domain (mathematical analysis) , combinatorics , function (biology) , metric (unit) , metric space , discrete mathematics , statistics , mathematical analysis , physics , operations management , evolutionary biology , acoustics , economics , biology
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 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.