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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2405.01465 |
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| _version_ | 1866917655809622016 |
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| author | Rached, Nadhir Ben Hoel, Håkon Meo, Johannes Vincent |
| author_facet | Rached, Nadhir Ben Hoel, Håkon Meo, Johannes Vincent |
| contents | We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01465 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables Rached, Nadhir Ben Hoel, Håkon Meo, Johannes Vincent Computation Numerical Analysis 60E05 (Primary) 65G50, 90-04 (Secondary) We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings. |
| title | A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables |
| topic | Computation Numerical Analysis 60E05 (Primary) 65G50, 90-04 (Secondary) |
| url | https://arxiv.org/abs/2405.01465 |