This paper deals with the problem of estimating the level sets
L(c) = {F(x) ≥ c},
with c ∈ (0,1), of an unknown distribution function F on ℝ+2. A plug-in
approach is followed. That is, given a consistent estimator
Fn of F, we estimate
L(c) by
Ln(c) = {Fn(x) ≥ c}.
In our setting, non-compactness property is a priori required for the
level sets to estimate. We state consistency results with respect to the Hausdorff
distance and the volume of the symmetric difference. Our results are motivated by
applications in multivariate risk theory. In particular we propose a new bivariate version
of the conditional tail expectation by conditioning the two-dimensional random vector to
be in the level set L(c). We also present simulated and
real examples which illustrate our theoretical results.