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Plug-in estimation of level sets in a non-compact setting withapplications in multivariate risk theory
Published online by Cambridge University Press: 08 February 2013
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This paper deals with the problem of estimating the level setsL(c) = {F(x) ≥ c},with c ∈ (0,1), of an unknown distribution function F on ℝ+2. A plug-inapproach is followed. That is, given a consistent estimatorFn of F, we estimateL(c) byLn(c) = {Fn(x) ≥ c}.In our setting, non-compactness property is a priori required for thelevel sets to estimate. We state consistency results with respect to the Hausdorffdistance and the volume of the symmetric difference. Our results are motivated byapplications in multivariate risk theory. In particular we propose a new bivariate versionof the conditional tail expectation by conditioning the two-dimensional random vector tobe in the level set L(c). We also present simulated andreal examples which illustrate our theoretical results.
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