A Threestep Method for Choosing Bootstrap Repetitions

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A Three-step Method for Choosing Bootstrap
Repetitions

• By Donald W. K. Andrews and Moshe Buchinsky

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Motivation

• B is determined in an ad hoc manner one can
  obtain a different answer from the same data
  merely by using different simulation draws if B
  is chosen to be too small.
• Intensive computational cost if B is too large

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Goal of this paper

• To provide methods for determining B to attain
  specified levels of accuracy for bootstrap
  standard error estimates, confidence intervals,
  confidence regions, hypothesis tests, and bias
  correction.

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Notes

• 1. The measure of accuracy employed is the
  percentage deviation of the bootstrap quantity of
  interest based on B repetitions from the ideal
  bootstrap quantity, for which B .
• 2. Applicable whenever the bootstrap samples are
  simulated to be iid across different bootstrap
  samples.

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Background on bootstrap
1.
2. and
3. and
4. and
5. and
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Measuring Accuracy
1. Measure
2. Given
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• 1 to be a bound of percentage
  deviation of . One can
  specify it according to the purpose of research
  or tolerance of accuracy. (ad hoc manner).
• 2. is the analogous to significance level.
  
• 3. Basically, one can attain some pre-specified
  significance level and a certain accuracy level
  using an optimal number of repetitons of
  bootstrap.

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Central result
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Corresponding bootstrap values
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Continue
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What can we use it?
Z is standard normal.
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An example of standard error case
and
0.95
B100001.9620.5/(102)200
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3-step method
1.
2.
3.
4.
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Notes

• 1. Can iterate step 1 and step 2. (not much
  improvement though)
• 2. For more than one quantities of interest,
  pretty much same. (conservative rule)
• 3. Do not throw away repetitions.

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Asymptotic justification
Note that when pdb goes to zero, 3-step method
attains the pre-specified accuracy level
precisely.
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Sketch of proof (Standard error case)

• step1. It is true as
  long as it is a
• consistent estimator.
• step2.
  
• Proof

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• By central limit theorem,

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• Using delta method

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Monte Carlo

• Regression model
• n25.
• 250 samples (with 3 different error
  distributions normal, t, chi-square)
• Ideal bootstrap number 250000 (benchmark)
• 3-step method (2000 Monte Carlo simulations)

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Results (normal)
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Results (t)
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• Chi-square results are similar to t-table.
• 3-step method attains the pre-specified accuracy
  level. (a little conservative)
• Good unconditional coverage probability.

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The End

• Thank you!