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

Nonparametric Methods II

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Henry Horng-Shing LuInstitute of Statistics

National Chiao Tung [email protected]

http://tigpbp.iis.sinica.edu.tw/courses.htm

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): 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.

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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_(statistics)

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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.


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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.


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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.


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Bootstrap Confidence Intervals1. A Simple Method2. 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 Distribution

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2 21 2

2

/2 /2

1

/2 /2

, , ..., ~ ( , ), is known.

ˆˆ ~ ( , ), ~ (0, 1).

/ˆ

( ) 1/

where 1

ˆ ˆ( / / ) 1

iid

n

LCL UCL

X X X N

X N Z Nn n

P z zn

Z

P z n z n

Asymptotic CI for The MLE More generally,

i.i.d.Let is MLE, then

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

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1 2, , , nX X X F x

ˆ

0,1ˆ. .

nX N

s e

ˆ

ˆ ˆ

ˆ1 as

ˆ ˆ 1 as

P Z Z n

P Z Z n

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.

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Simple Methods (1) Theorem in Gill (1989):

Under regular conditions,

Want

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*1

ˆ ,

ˆ ˆ , , .

on

on n

n F d F B F

n X X d F B F

1P LCL UCL

Simple Methods (2)Note that

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* * **

12 2

* * *

12 2

* *

12 2

ˆ ˆ ˆ ˆ ˆ ˆ1

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ 2 2

P

P

P

P LCL UCL

An Example by The Simple Method (1)

Resampling with replacement from

Repeat times,we can get .

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11 2 101

11 2 101 51

1, , , ~ , median

2

1ˆ, 2

iid

n

X X X N F

X X X F X

1 101, ,X X

* * *1 2 101

* * 1 *51

1ˆ2n

X X X

F X

1000B

* * *1 2 1000

ˆ ˆ ˆ

An Example by The Simple Method (2)

is an approximate confidence interval for .

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*(1) *

(1000)*(25) *

(975)

95%

* * ** 25 975

* * ** 25 975

* * * *25 975 975 25

* *975 25

ˆ ˆ ˆ 1

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 2

ˆ ˆ ˆ ˆ2 , 2

P

P

P P

LCL UCL

100 1 %

Flowchart of The Simple Method

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*2x

*Bx

*(2)

1 2ˆ ( , , ..., ) ( )ndata x x x s x x

*1x

resample B times

*(1)

1 2[( 1) / 2], [( 1)(1 / 2)]v B v B

2 1

* *( ) ( )

ˆ ˆ ˆ ˆ2 , 2v vLCL UCL

*( )ˆB*

(2)

get resample statistics and then sort them * *b bs x

confidence interval 100 1 %

The Simple Method by R (1)

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The Simple Method by R (2) Example

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The Simple Method by C (1)

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resample B times:

* *ˆ ( )b bmean x

*bx

ˆ ( ) ( )s x mean x


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calculate v1, v2 and interval


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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) Suppose

Then

Assume that there exists an unbiased and (monotonly) increasing function such that .

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ˆ ~Y H

~H Y U

1 1~ ~ 0,1H Y U N

g

ˆ 0,1g g N

The Percentile Method (3) If , then

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ˆ 0,1g g N *ˆ ˆ 0,1g g N

1

**

* 1 ** 1 1

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1 *1 1 1

ˆ ˆ 1

ˆ ˆ ˆ ˆ and

ˆ

ˆ Notice: for 0,1

ˆ ˆ ˆ and

B

B

P g g z

P g g z

P g g z

P g g z z z N

P g g z

The Percentile Method (4) Similarly,

and

Summary of the percentile method:

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*

1 1ˆ 1

BP

* *

1 /2 1 1 /2ˆ ˆ 1

B BP

*

1

*

1 1

* *

1 /2 1 1 /2

ˆ 1

ˆ 1

ˆ ˆ 1

B

B

B B

P

P

P

Flowchart of The Percentile Method

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*2x

*Bx

*(2)

1 2ˆ ( , , ..., ) ( )ndata x x x s x x

*1x

resample B times

*(1)

1 2[( 1) / 2], [( 1)(1 / 2)]v B v B

*( )ˆB*

(2)



1 2

* *( ) ( )ˆ ˆ,v vLCL UCL

The Percentile Method by R (1)

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The Percentile Method by R (2) Example

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The Percentile Method by C (1)

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calculate v1, v2 and interval


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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

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*2x

*Bx

*(2)

1 2ˆ ( , , ..., ) ( )ndata x x x s x x

*1x

resample B times

*(1)

1 0 1 2 02 , 2v z z v z z

*( )ˆB*

(2)



1 2

* *

1 1ˆ ˆ,B v B v

LCL UCL

0estimate z

1( ) z

1 *0

1

1 ˆ ˆestimate by 1B

bb

zB

The BC Percentile Method by R (1)

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The BC Percentile Method by R (2) Example

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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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1 1

1 2

*

** 0

* 1 ** 0 *

0

01

0

1 *0

1 1

ˆ ˆ1

ˆ1

ˆ ˆ ˆ ˆ1

ˆ1

1

ˆ

1

ˆ ˆ ˆ1

ˆ ˆB

g gP U z z

a g

P g g a g z z P

g gP U z z

a g

g z zP g

a z z

P g g a g z z P

*

The BCa Percentile Method (3) Similarly,

and

and

and44

2

*

1 1ˆ 1

BP

1 2

* *

1 1 1 1ˆ ˆ 1 2

B BP

11 1? 1 P z

1

0

00

ˆˆ ˆ1

1

g z zg a g z z

a z z

0

1 00

11

z zP Z z

a z z

1

00

01

z zz z

a z z

The BCa Percentile Method (4) Similarly,

and

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1 0

2 01 0

11

z zP Z z

a z z

0

* ** *

*

* 0 0 0

?

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ1 1

z

P P g g

g g g gP z z z

a g a g

1 *0 *

ˆ ˆz P 1 *0

1

1 ˆ ˆˆ 1B

bb

zB

The BCa Percentile Method (5)

where

and

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3

13/2

2

1

?

ˆ ˆ

ˆˆ ˆ6

n

ii

Jackn

ii

a

a

1, 1ˆ , , , ,n i i ni F X X X

1

1ˆ ˆn

iin

Flowchart of The BCa Percentile Method

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*2x

*Bx

*(2)

1 2ˆ ( , , ..., ) ( )ndata x x x s x x

*1x

resample B times

*(1) *

( )ˆB*

(2)



0estimate ,z a

1( ) z

1 *0

1

1 ˆ ˆestimate by 1 and by JackknifeB

bb

z aB

/ 2 0 1 / 2 01 0 2 0

/ 2 0 1 / 2 0

1 ( ), 1 ( )1 ( ) 1 ( )

z z z zz z

a z z a z z

1 2

* *

1 1 1 1ˆ ˆ,B B

LCL UCL

The BCa Percentile Method by R (1)

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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 (4) Example

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The BCa Percentile Method by C (1)

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

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