Surface integrals on density level sets often appear in asymptotic results of nonparametric level set estimation, such as for confidence regions and bandwidth selection. Also surface integrals can be used to describe the shape of level sets. Assuming the integrands are known, we consider three estimators of the surface integrals on density level sets, one as a direct plug-in estimator, and the other two based on different neighborhoods of level sets. For all the three estimators, we derive the rates of convergence and asymptotic distributions.
"Nonparametric estimation of surface integrals on level sets." Bernoulli 27 (1) 155 - 191, February 2021. https://doi.org/10.3150/20-BEJ1232