The Empirical Beta Copula

• Activity Number: 178
• Type: Contributed
• Date/Time: Monday, August 10, 2015 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Hideatsu Tsukahara* and Johan Segers and Masaaki Sibuya and Nathan Uyttendaele
• Companies: and Université catholique de Louvain and Keio University and Université catholique de Louvain
• Keywords: empirical copula ; smoothing ; Beta distribution

Abstract

The pseudo-observations are defined to be the empirical probability integral transforms of the data. They are discrete in nature, as can be seen from the fact that they equal simple functions of the coordinate-wise ranks. The empirical copula, which is defined as the empirical distribution functions of the pseudo-observations, is itself not a copula, and a piecewise constant function. Therefore, in view of practical applications, one might hope that there is an advantage of smoothing to improve finite-samples performance and also the accuracy of resampling schemes. We suggest a new version of smoothed empirical copula, called the empirical Beta copula, which has the advantage of not requiring any smoothing parameter. Also it is extremely simple to simulate a sample from the empirical Beta copula. We show the the empirical Beta copula is consistent whenever the empirical copula is, and its first order asymptotics is the same as the empirical copula. Moreover we study its finite-sample properties with Monte Carlo simulation. It is found that in all cases, the empirical Beta copula outperforms the alternative estimators in terms of the bias and integrated mean squared error.