دانلود رایگان مقاله برآورد خطای مرتبه اول و دوم در ادغام مونت کارلو

Abstract

In Monte Carlo integration an accurate and reliable determination of the numerical integration error is essential. We point out the need for an independent estimate of the error on this error, for which we present an unbiased estimator. In contrast to the usual (first-order) error estimator, this second-order estimator can be shown to be not necessarily positive in an actual Monte Carlo computation. We propose an alternative and indicate how this can be computed in linear time without risk of large rounding errors. In addition, we comment on the relatively very slow convergence of the second-order error estimate.

Conclusions

We have argued that the current practice of MC integration, resulting in a report on the integral estimate and its error estimate, should always be accompanied by a second-order error estimate, if only to validate the assignment of confidence levels to the result (which can be, for instance, crucial in comparing the results of different MC calculations, which is good and common practice). We have presented the relevant estimators. A closer look at E4 shows potential positivity problems and we have emended this by defining an improved estimator Eˆ 4. We also point out that, on the one hand, the convergence of the second-order error, Eˆ 1/4 4 /E2 1/2 ∼ N −1/4 , rather than the ‘well-known’ E2 1/2 /E1 ∼ N −1/2 convergence of the error itself, and that on the other hand E2 satisfies its own version of the central-limit theorem. In addition, we have extended the methods of the Chan–Golub–Leveque algorithm [4] to allow for a numerically stable computation of not only E2 but Eˆ 4 as well.