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

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

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. 3

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.  (statistics) (statistics) 4

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.  (statistics) (statistics) 5

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.  (statistics) (statistics) 6

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.  (statistics) (statistics) 7

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.) 8

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

Normal Distribution 10

Asymptotic CI for The MLE  More generally, i.i.d. Let is MLE, then 11

Bootstrap Confidence Intervals  When n is not large, we can construct more precise confidence intervals by bootstrap methods for many statistics including the MLE and others. 12

Simple Methods (1)  Theorem in Gill (1989): Under regular conditions, Want 13

Simple Methods (2) Note that 14

An Example by The Simple Method (1)  Resampling with replacement from Repeat times, we can get. 15

An Example by The Simple Method (2) is an approximate confidence interval for. 16

Flowchart of The Simple Method 17 resample B times get resample statistics and then sort them confidence interval

The Simple Method by R (1) 18

The Simple Method by R (2)  Example 19

The Simple Method by C (1) 20 resample B times:

The Simple Method by C (2) 21 calculate v1, v2 and interval

The Simple Method by C (3) 22

The Simple Method by C (4) 23

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

2.1. The Percentile Method  Methodology  Flowchart  R codes  C codes 25

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.  at_080/pdf/moore14.pdf at_080/pdf/moore14.pdf 26

The Percentile Method (2)  Suppose Then Assume that there exists an unbiased and (monotonly) increasing function such that. 27

The Percentile Method (3)  If, then 28

The Percentile Method (4)  Similarly, and Summary of the percentile method: 29

Flowchart of The Percentile Method 30 resample B times get resample statistics and then sort them confidence interval

The Percentile Method by R (1) 31

The Percentile Method by R (2)  Example 32

The Percentile Method by C (1) 33 calculate v1, v2 and interval

The Percentile Method by C (2) 34

The Percentile Method by C (3) 35

2.2. The BC Percentile Method  Methodology  Flowchart  R code 36

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. 37

Flowchart of The BC Percentile Method 38 resample B times get resample statistics and then sort them confidence interval

The BC Percentile Method by R (1) 39

The BC Percentile Method by R (2)  Example 40

2.3. The BCa Percentile Method  Methodology  Flowchart  R code  C code 41

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.  at_080/pdf/moore14.pdf at_080/pdf/moore14.pdf 42

The BCa Percentile Method (2) 43

The BCa Percentile Method (3)  Similarly, and 44

The BCa Percentile Method (4)  Similarly, and 45

The BCa Percentile Method (5) where and 46

Flowchart of The BCa Percentile Method 47 resample B times get resample statistics and then sort them confidence interval

The BCa Percentile Method by R (1) 48 Step 1: Install the library of bootstrap in R. Step 2: If you want to check BCa, type “?bcanon”.

The BCa Percentile Method by R (2) 49

The BCa Percentile Method by R (3) 50

The BCa Percentile Method by R (4)  Example 51

The BCa Percentile Method by C (1) 52

The BCa Percentile Method by C (2) 53

The BCa Percentile Method by C (3) 54

The BCa Percentile Method by C (4) 55

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. 56