a neat (theoretical) Monte Carlo result

Mark Huber just arXived a short paper where he develops a Monte Carlo approach that bounds the probability of large errors

$\mathbb{P}(|\hat\mu_t-\mu|>\epsilon\mu) < 1/\delta$

by computing a lower bound on the sample size r and I wondered at the presence of μ in the bound as it indicates the approach is not translation invariant. One reason is that the standard deviation of the simulated random variables is bounded by cμ. Another reason is that Mark uses as its estimator the median

$\text{med}(S_1R_1,\ldots,S_tR_t)$

where the S’s are partial averages of sufficient length and the R’s are independent uniforms over (1-ε,1+ε): using those uniforms may improve the coverage of given intervals but it also means that the absolute scale of the error is multiplied by the scale of S, namely μ. I first thought that some a posteriori recentering could improve the bound but since this does not impact the variance of the simulated random variables, I doubt it is possible.

This entry was posted on December 19, 2014 at 12:14 am and is filed under Books, Statistics, University life with tags confidence sets, importance sampling, Monte Carlo integration, unbiased estimation, variance reduction. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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