Asymptotic confidence bands for copulas based on the transformation kernel estimator

TL;DR

This paper develops asymptotic confidence bands for a transformation kernel estimator of copulas, providing theoretical guarantees and simulations to validate the estimator's accuracy and bias behavior.

Contribution

It introduces asymptotic confidence bands for the transformation kernel estimator of copulas and proves a uniform law of the iterated logarithm under smoothness conditions.

Findings

Established asymptotic confidence bands for the estimator

Proved a uniform law of the iterated logarithm for the deviation

Showed the bias tends to zero uniformly at a precise rate

Abstract

In this paper we establish asymptotic simultaneous confidence bands for the transformation kernel estimator of copulas introduced in Omelka et al.(2009). To this aim, we prove a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation, under smoothness conditions on the copula function. We also study the bias, which tends asymptotically and uniformly to zero with the same precise rate. Some simulation experiments are finally provided to support our results