Estimation of Copula Density Using the Wavelet Transform

Abstract

This paper proposes a new method to estimate the copula density function using wavelet decomposition as a nonparametric method, to obtain more accurate results and address the issue of boundary effects that nonparametric estimation methods suffer from. The wavelet method is an automatic method for dealing with boundary effects because it does not take into Consideration whether the time series is stationary or nonstationary. To estimate the copula density function, simulation was used to generate data using five different copula functions, such as Gaussian, Frank, Tawn, Rotation Tawn, and Joe copulas. With five different sample sizes at three positive correlation levels based on multiresolution. The results showed that in estimating the copula density function using the wavelet method when the correlation level  = 0.7, the Gaussian copula ranked first, followed by the Frank copula, and the Joe copula ranked last. In the case of medium and weak correlation, the Tawn copula was in first place, followed by the Rotation Tawn copula, while Gaussian copula came in last place depending on the measures (Root Mean Square Error, Akiake Information Criteria, and Logarithm likelihood criteria). The real copula functions are shown through drawing (Contour plot) and (3D plot). In addition to the smoothing shapes for each of them using the wavelet method, it is clear from the circular shapes that the distribution of observations of the copula function estimated with the wavelet method was accurate at the edges, while it was less accurate at the center for Gaussian and Tawn functions.