A Mathematics Dissertation Defense by John Kirby is Thursday, Nov. 13, from 11:30 a.m. to 1:30 p.m. in Morton Hall 320.
Kirby’s presentation is “Multiscale Copula Sampling Series and Applications” (directed by Dr. Xiaoping Shen).
Abstract: Copulas are multivariate distribution functions with uniform marginals. They can be used as a flexible representation of the multivariate distribution measurement to describe the dependence structure between random variables such as the dependence between risk factors. In recent years, copulas have become a promising solution in modeling multivariate data from hazard insurance, risk management, and analysis extremes in nature, to name a few.
Many challenging problems occur in copula theory and its applications.
In this dissertation, we focus on
- Introducing a new family of Archimedean copulas. The copula family is generated via a wavelet scaling function—the raised cosine wavelet;
- Approximating copula using raised cosine wavelets. Raised cosine wavelets are a family of bandlimited wavelets with closed form. The decay rate is faster than other popular (continuous) wavelets, such as the Shannon wavelets. Raised cosine wavelets series approximation for copula density is constructed;
- Overcoming the Gibbs phenomenon in wavelet copula expansion. To combat the common problem raised in wavelet approximation in copula, the sampling function associated with the Daubechies (db2) wavelet is used to define multiscale wavelet copula via periodization process to meet the necessary condition of copulas. This approach is based on an observation that the sampling function for the Daubechies (db2) wavelet has a peculiar property – positivity. We show that the new sampling wavelet copula does not exhibit Gibbs phenomenon. As a side result, we also successfully illustrated the existence of Gibbs phenomenon in wavelet copula approximations.
Finally, we apply the new multiscale copulas in the data analysis of risk management – insurance data modeling.
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