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Chapter 6: Monte Carlo Methods for Inferential Statistics 219

ias
BB iasias
Estimate
BootstrapEstimateof
Bootstrap
Bootstrap Estimate of B B ias
BootstrapEstimateofof
The standard error of an estimate is one measure of its performance. Bias is
another quantity that measures the statistical accuracy of an estimate. From
Chapter 3, the bias is defined as the difference between the expected value of
the statistic and the parameter,
bias T() = ET[] – θ . (6.16)
The expectation in Equation 6.16 is taken with respect to the true distribution
F. To get the bootstrap estimate of bias, we use the empirical distribution F ˆ
as before. We resample from the empirical distribution and calculate the sta-
tistic using each bootstrap resample, yielding the bootstrap replicates θ ˆ *b . We
use these to estimate the bias from the following:

ˆ *
ˆ
bias B = θ – ˆ , θ (6.17)
where θ ˆ * is given by the mean of the bootstrap replicates (Equation 6.15).
Presumably, one is interested in the bias in order to correct for it. The bias-
corrected estimator is given by

) ˆ
ˆ
θ = θ – bias B . (6.18)
Using Equation 6.17 in Equation 6.18, we have

) *
ˆ
θ = 2θ – θ ˆ . (6.19)
More bootstrap samples are needed to estimate the bias, than are required
to estimate the standard error. Efron and Tibshirani [1993] recommend that
B ≥ 400 .
ˆ
θ
It is useful to have an estimate of the bias for , but caution should be used
when correcting for the bias. Equation 6.19 will hopefully yield a less biased
estimate, but it could turn out that θ ) will have a larger variation or standard
error. It is recommended that if the estimated bias is small relative to the esti-
mate of standard error (both of which can be estimated using the bootstrap
method), then the analyst should not correct for the bias [Efron and Tibshi-
rani, 1993]. However, if this is not the case, then perhaps some other, less
biased, estimator should be used to estimate the parameter . θ

PROCEDURE - BOOTSTRAP ESTIMATE OF THE BIAS

ˆ
1. Given a random sample, x = ( x … x, , n ) , calculate the statistic . θ
1

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