The bootstrap method for Monte Carlo integration inference

Abstract

In this paper, the use of bootstrap method with Monte Carlo integration is introduced for one dimension. This approach is based on generating observations from a known distribution for the bootstrap samples, then apply the Monte Carlo method on each bootstrap sample to estimate the integral of interest. The empirical distribution, or the bootstrap distribution, of the estimation results can be used as a good proxy for the distribution of the integral of interest. Based on the bootstrap distribution, the standard error of the estimate of the integral of interest can be derived. Also, the percentile and Normal confidence intervals with confidence level (1-α)% can be derived as well. The bootstrap method with Monte Carlo integration is easy to implement and straightforward to provide well results. Moreover, it provides small variance for the estimate of the integral of interest. Four examples with different functions and different domains are used to present the performance of the proposed method. From the study, we find that the method provides nearly identical results for the standard errors, regardless of the distributions used for generating observations for the bootstrap samples.

Keywords

• Parametric bootstrap method
• Monte Carlo integration
• Percentile confidence interval
• Normal confidence interval
• Standard error
• Simulation