Abstract
We propose a new goodness-of-fit test for copulas using the collision test for pseudorandom number generators and Voronoi diagrams generated by low-discrepancy sequences. We provide an error bound for numerical integration that involves number of collisions when the unit cube is partitioned via Voronoi cells, and present an example from option pricing. We investigate the accuracy of the goodness-of-fit test numerically, and compare it with three tests in the literature. The numerical results suggest the new test excels in computationally demanding scenarios when the sample size is large or computing the copula is expensive, and provides sufficient accuracy in computing times that are faster by factors of thousands than the tests with comparable accuracy.
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Notes
One could also consider finite point sets such as lattice rules.
A simple modification of the original implementation of the ziggurat method given in Marsaglia and Tsang (2000) corrects the problem.
Other low-discrepancy sequences such as Sobol’ and Halton, as well as (t, m, s)-nets and lattice rules can be used in constructing Voronoi diagrams.
R is the average rejection rate over 100 simulated data, and t is the computing time for only one simulation.
The computer codes are implemented in MATLAB R2015a on a computer with Intel 9700K, 4.6 GHz, 8 cores CPU, and 2.8 gigabyte memory.
We implemented methods \({\mathcal {A}}_1, {\mathcal {A}}_2, {\mathcal {A}}_3\) according to the algorithms given by Berg (2009). To make \({\mathcal {A}}_2\) and \({\mathcal {A}}_3\) run faster, we lowered the number of bootstrapping repetitions from 10,000 to 1000 while estimating the p value.
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Acknowledgements
We thank Dr. Phil Bowers and Dr. John Bowers for helpful discussions on Voronoi diagrams.
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Chen, Y., Ökten, G. A goodness-of-fit test for copulas based on the collision test. Stat Papers 63, 1369–1385 (2022). https://doi.org/10.1007/s00362-021-01277-6
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DOI: https://doi.org/10.1007/s00362-021-01277-6