1 Nonparametric Methods II Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University

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.

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 intervals for complex estimators of complex parameters of the distribution, such as percentile points, proportions, odds ratio, and correlation coefficients.

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.

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.

8 Bootstrap Confidence Intervals 1. A Simple Method 2. 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

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

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

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

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24 2. Transformation Methods  2.1. The Percentile Method  2.2. The BC Percentile Method  2.3. The BCa Percentile Method

The Percentile Method  Methodology  Flowchart  R codes  C codes

26 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.

27 The Percentile Method (2)

28 The Percentile Method (3)

29 The Percentile Method (4)

30 Flowchart of The Percentile Method resample B times

31 The Percentile Method by R

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

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The BC Percentile Method  Methodology  Flowchart  R code

38 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.

39 Flowchart of The BC Percentile Method resample B times

40 The BC Percentile Method by R

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The BCa Percentile Method  Methodology  Flowchart  R code  C code

43 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.

44 The BCa Percentile Method (2)

45 The BCa Percentile Method (3)

46 The BCa Percentile Method (4)

47 The BCa Percentile Method (5)

48 Flowchart of The BCa Percentile Method resample B times

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

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51 The BCa Percentile Method by R

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53 The BCa Percentile Method by C

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59 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. 59