Nonparametric Methods II

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Nonparametric Methods II

• Henry Horng-Shing Lu
• Institute of Statistics
• National Chiao Tung University
• hslu_at_stat.nctu.edu.tw
• http//tigpbp.iis.sinica.edu.tw/courses.htm

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PART 3 Statistical Inference by Bootstrap Methods

• References
• Pros and Cons
• Bootstrap Confidence Intervals
• Bootstrap Tests

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References

• Efron, B. (1979). "Bootstrap Methods Another
  Look at the Jackknife". The Annals of Statistics
  7 (1) 126. 
• Efron, B. Tibshirani, R. (1993). An Introduction
  to the Bootstrap. Chapman Hall/CRC.
• Chernick, M. R. (1999). Bootstrap Methods, A
  practitioner's guide. Wiley Series in Probability
  and Statistics.

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Pros (1)

• In statistics, bootstrapping is a modern,
  computer-intensive, general purpose approach to
  statistical inference, falling within a broader
  class of re-sampling methods.

http//en.wikipedia.org/wiki/Bootstrapping_(statis
tics)
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Pros (2)

• The advantage of bootstrapping over analytical
  method is its great simplicity - it is
  straightforward to apply the bootstrap to derive
  estimates of standard errors and confidence
  intervals for complex estimators of complex
  parameters of the distribution, such as
  percentile points, proportions, odds ratio, and
  correlation coefficients.

http//en.wikipedia.org/wiki/Bootstrapping_(statis
tics)
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Cons

• The disadvantage of bootstrapping is that while
  (under some conditions) it is asymptotically
  consistent, it does not provide general finite
  sample guarantees, and has a tendency to be
  overly optimistic.

http//en.wikipedia.org/wiki/Bootstrapping_(statis
tics)
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How many bootstrap samples is enough?

• As a general guideline, 1000 samples is often
  enough for a first look. However, if the results
  really matter, as many samples as is reasonable
  given available computing power and time should
  be used.

http//en.wikipedia.org/wiki/Bootstrapping_(statis
tics)
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Bootstrap Confidence Intervals

• A Simple Method
• Transformation Methods
• 2.1. The Percentile Method
• 2.2. The BC Percentile Method
• 2.3. The BCa Percentile Method
• 2.4. The ABC Method (See the book An
  Introduction to the Bootstrap.)

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1. A Simple Method

• Methodology
• Flowchart
• R codes
• C codes

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Normal Distributions
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Asymptotic C. I. for The MLE
http//en.wikipedia.org/wiki/Pivotal_quantity
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Bootstrap Confidence Intervals
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Simple Methods
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An Example by The Simple Method (1)
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An Example by The Simple Method (2)
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Flowchart of The Simple Method
resample B times
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The Simple Method by R
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The Simple Method by C (1)
resample B times
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The Simple Method by C (2)
calculate v1, v2
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2. Transformation Methods

• 2.1. The Percentile Method
• 2.2. The BC Percentile Method
• 2.3. The BCa Percentile Method

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2.1. The Percentile Method

• Methodology
• Flowchart
• R codes
• C codes

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The Percentile Method (1)

• The interval between the 2.5 and 97.5
  percentiles of the bootstrap distribution of a
  statistic is a 95 bootstrap percentile
  confidence interval for the corresponding
  parameter. Use this method when the bootstrap
  estimate of bias is small.

http//bcs.whfreeman.com/ips5e/content/cat_080/pdf
/moore14.pdf
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The Percentile Method (2)
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The Percentile Method (3)
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The Percentile Method (4)
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Flowchart of The Percentile Method
resample B times
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The Percentile Method by R
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The Percentile Method by C
resample B times
calculate v1, v2
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2.2. The BC Percentile Method

• Methodology
• Flowchart
• R code

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The BC Percentile Method

• Stands for the bias-corrected percentile method.
  This is a special case of the BCa percentile
  method which will be explained more later.

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Flowchart of The BC Percentile Method
resample B times
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The BC Percentile Method by R
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2.3. The BCa Percentile Method

• Methodology
• Flowchart
• R code
• C code

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The BCa Percentile Method (1)

• The bootstrap bias-corrected accelerated (BCa)
  interval is a modification of the percentile
  method that adjusts the percentiles to correct
  for bias and skewness.

http//bcs.whfreeman.com/ips5e/content/cat_080/pdf
/moore14.pdf
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The BCa Percentile Method (2)
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The BCa Percentile Method (3)
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The BCa Percentile Method (4)
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The BCa Percentile Method (5)
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Flowchart of The BCa Percentile Method
resample B times
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Step 1 Install the library of
bootstrap in R.
Step 2 If you want to check BCa, type
?bcanon.
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The BCa Percentile Method by R
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The BCa Percentile Method by C
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Exercises

• Write your own programs similar to those examples
  presented in this talk.
• Write programs for those examples mentioned at
  the reference web pages.
• Write programs for the other examples that you
  know.
• Prove those theoretical statements in this talk.

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