Uniform in bandwidth consistency for the transformation kernel estimator of copulas

TL;DR

This paper proves uniform in bandwidth consistency for a copula estimator, establishing theoretical properties, bias convergence, and practical bandwidth selection, supported by simulation experiments demonstrating finite sample performance.

Contribution

It introduces a uniform in bandwidth law of the iterated logarithm and a practical bandwidth selection method for the transformation kernel estimator of copulas.

Findings

Uniform in bandwidth law of the iterated logarithm established

Bias of the estimator converges uniformly to zero as sample size increases

Simulation experiments confirm good finite sample performance

Abstract

In this paper we establish the uniform in bandwidth consistency for the transformation kernel estimator of copulas introduced in [Omelka et al.(2009)]. To this end, we first prove a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We then show that, as n goes to infinity, the bias of the estimator converges to zero uniformly in the bandwidth h, varying over a suitable interval. A practical method of selecting the optimal bandwidth is also presented. Finally, we make conclusive simulation experiments showing the performance of the estimator in finite samples.