Inverse Sampling for Nonasymptotic Sequential Estimation of Bounded Variable Means

Abstract

In this article, we consider the nonasymptotic sequential estimation of means of random variables bounded in between zero and one. We have rigorously demonstrated that, in order to guarantee a prescribed relative precision and confidence level, it suffices to continue sampling until the sample sum is no less than a certain bound and then take the sample mean as an estimate for the mean of the bounded random variable. We have developed an explicit formula and a bisection search method for the determination of such bound of sample sum, without any knowledge of the bounded variable. Moreover, we have derived bounds for the distribution of sample size. In the special case of Bernoulli random variables, we have established analytical and numerical methods to further reduce the bound of sample sum and thus improve the efficiency of sampling.

Keywords:

• Bounded variable
• Inverse sampling
• Monte Carlo estimation
• Nonasymptotic method
• Sequential estimation
• Uncertainty

Subject Classifications:

• 62L12
• 62D05

ACKNOWLEDGMENTS

The author would like to thank the Editor, Associate Editor, and referees for their time, effort, and comments in reviewing this article. Preliminary results from this article were posted at http://arxiv.org/abs/0711.2801 on November 18, 2007, and appeared in the Proceedings of the SPIE Conference, Orlando, FL, April 5–9, 2010.

Notes

Recommended by Nitis Mukhopadhyay