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Nonparametric Methods II

Nonparametric Methods II. Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University hslu@stat.nctu.edu.tw http://tigpbp.iis.sinica.edu.tw/courses.htm. PART 3: Statistical Inference by Bootstrap Methods. References Pros and Cons Bootstrap Confidence Intervals

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Nonparametric Methods II

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  1. Nonparametric Methods II Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University hslu@stat.nctu.edu.tw http://tigpbp.iis.sinica.edu.tw/courses.htm

  2. PART 3: Statistical Inference by Bootstrap Methods • References • Pros and Cons • Bootstrap Confidence Intervals • Bootstrap Tests

  3. References • Efron, B. (1979). "Bootstrap Methods: Another Look at the Jackknife". The Annals of Statistics 7 (1): 1–26. • 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.

  4. 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_(statistics)

  5. 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 intervalsfor complex estimators of complex parameters of the distribution, such as percentile points, proportions, odds ratio, and correlation coefficients. http://en.wikipedia.org/wiki/Bootstrapping_(statistics)

  6. 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_(statistics)

  7. 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_(statistics)

  8. 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.)

  9. 1. A Simple Method • Methodology • Flowchart • R codes • C codes

  10. Normal Distributions

  11. Asymptotic C. I. for The MLE http://en.wikipedia.org/wiki/Pivotal_quantity

  12. Bootstrap Confidence Intervals

  13. Simple Methods

  14. An Example by The Simple Method (1)

  15. An Example by The Simple Method (2)

  16. Flowchart of The Simple Method resample B times

  17. The Simple Method by R

  18. The Simple Method by C (1) resample B times:

  19. The Simple Method by C (2) calculate v1, v2

  20. 2. Transformation Methods • 2.1. The Percentile Method • 2.2. The BC Percentile Method • 2.3. The BCa Percentile Method

  21. 2.1. The Percentile Method • Methodology • Flowchart • R codes • C codes

  22. The Percentile Method (1) • The interval between the 2.5% and 97.5% percentiles of the bootstrapdistribution of a statistic is a 95%bootstrap percentile confidenceinterval for the corresponding parameter. Use this method when thebootstrap estimate of bias is small. http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf

  23. The Percentile Method (2)

  24. The Percentile Method (3)

  25. The Percentile Method (4)

  26. Flowchart of The Percentile Method resample B times

  27. The Percentile Method by R

  28. The Percentile Method by C resample B times: calculate v1, v2

  29. 2.2. The BC Percentile Method • Methodology • Flowchart • R code

  30. 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.

  31. Flowchart of The BC Percentile Method resample B times

  32. The BC Percentile Method by R

  33. 2.3. The BCa Percentile Method • Methodology • Flowchart • R code • C code

  34. The BCa Percentile Method (1) • The bootstrap bias-corrected accelerated (BCa) intervalis 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

  35. The BCa Percentile Method (2)

  36. The BCa Percentile Method (3)

  37. The BCa Percentile Method (4)

  38. The BCa Percentile Method (5)

  39. Flowchart of The BCa Percentile Method resample B times

  40. Step 1: Install the library of bootstrap in R. Step 2: If you want to check BCa, type “?bcanon”.

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