A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables
<|reference_start|>A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables: 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.<|reference_end|>
@article{rached2024a,
    title={A Fast and Accurate Numerical Method for the Left Tail of Sums of
  Independent Random Variables},
    author={Nadhir Ben Rached, H{aa}kon Hoel and Johannes Vincent Meo},
    journal={arXiv preprint arXiv:2405.01465},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.01465},
    primaryClass={stat.CO cs.NA math.NA}
}