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Independent RQMC replicates estimate randomization variance

Statement

Fix the observed data and an integrable RQMC target $I = I (f)$ . Suppose one full-set randomization produces an estimator $\hat{I}$ with $E [\hat{I}] = I$ and finite variance $τ^{2}$ . For $R \geq 2$ independent full-set randomizations, write their IID estimates as ${\hat{I}}_{1}, \dots, {\hat{I}}_{R}$ and their mean as ${\overline{I}}_{R} = R^{- 1} \sum_{r = 1}^{R} {\hat{I}}_{r}$ . Then ${\hat{V}}_{R} = \frac{1}{R (R - 1)} \sum_{r = 1}^{R} ({\hat{I}}_{r} - {\overline{I}}_{R})^{2}$ is unbiased for $Var ({\overline{I}}_{R}) = τ^{2} / R$ . This variance is over computation randomization conditional on the fixed data and rule; it does not include data-sampling variation.

Facts & Assumptions

Given: The fixed target, an RQMC estimator with mean $I$ and finite variance, and $R \geq 2$ independent full-set randomizations.

[F1]

Randomized quasi-Monte Carlo point sets and estimators: An independent full-set randomization is an independent copy of the entire point-array vector.

[F2]

Marginal uniformity makes an RQMC rule unbiased: A single marginally uniform rule has expectation $I (f)$ for every Borel integrable $f$ .

[F3]

Unbiased variance estimate from independent computation replicates: For IID real $Z_{1}, \dots, Z_{R}$ with finite variance, $[R (R - 1)]^{- 1} \sum_{r} (Z_{r} - {\overline{Z}}_{R})^{2}$ is unbiased for $Var ({\overline{Z}}_{R})$ .

[F4]

Moments, variance, and covariance on a probability space: Variance is $E [(Z - E Z)^{2}]$ for a square-integrable real variable.

Proof

technique · direct

Given: The hypotheses in the Statement.

1.1F1F2F3F4given

By [F1], independent full-set randomizations give independent estimates with the same law. By [F2] their common mean is $I$ , and the stated finite variance supplies the moment hypothesis of [F3].

2.1F3givenalgebra∎

Apply [F3] to $Z_{r} = {\hat{I}}_{r}$ . It gives $E [{\hat{V}}_{R}] = Var ({\overline{I}}_{R})$ . Independence and the common variance $τ^{2}$ give $Var ({\overline{I}}_{R}) = R^{- 2} \sum_{r} Var ({\hat{I}}_{r}) = τ^{2} / R$ . Since the data are fixed in the probability law of these replicates, this variance concerns randomization only.

Depends on

Used by

Dependency tree · two levels

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Sources

Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §17.1 (standard reference, not scraped)
Pierre L’Ecuyer and Christiane Lemieux, Recent Advances in Randomized Quasi-Monte Carlo Methods, §5 (standard reference, not scraped)