Nonparametric Methods II

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# Nonparametric Methods II - PowerPoint PPT Presentation

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

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
• Bootstrap Tests
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.
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)

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)

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)

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)

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.)

1. A Simple Method
• Methodology
• Flowchart
• R codes
• C codes
Asymptotic C. I. for The MLE

http://en.wikipedia.org/wiki/Pivotal_quantity

The Simple Method by C (1)

resample B times:

2. Transformation Methods
• 2.1. The Percentile Method
• 2.2. The BC Percentile Method
• 2.3. The BCa Percentile Method
2.1. The Percentile Method
• Methodology
• Flowchart
• R codes
• C codes
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

The Percentile Method by C

resample B times:

calculate v1, v2

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

Step 1: Install the library

of bootstrap in R.

Step 2: If you want to check

BCa, type “?bcanon”.

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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