Nonparametric Methods II

Transcription

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

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

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

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. 4

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. 5

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. 6

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. 7

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.) 8

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

10 Normal Distributions iid 2 2 1, 2,..., n ~ ( μ, σ ), σ is known. X X X N 2 ˆ ˆ σ θ θ θ = X ~ N( θ, ), Z = ~ N(0, 1). n σ / n ˆ θ θ P z z where Z σ / n P( ˆ θ z ˆ α/2 σ / n θ θ + zα/2σ / n) = 1 α. 1 ( α/2 α/2 ) = 1 α, α /2 =Φ (1 /2) LCL UCL α 10

11 Asymptotic C. I. for The MLE More generally, X, X,..., X ~ F ( x). 1 2 iid Let ˆ θ = MLE, then Pivot n θ ˆ θ θ = N(0, 1)...( ˆ n se θ ) ˆ θ θ n P( z z ) 1 α. α/2 α/2 σ ˆ θ P( ˆ θ z σ θ ˆ θ + z σ ) 1 α α/2 ˆ θ α/2 ˆ θ 11

12 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

13 Simple Methods Theorem in Gill (1989): Under regular conditions, n( ˆ o θ θ( F)) dθ( F) B F, n ˆ θ ˆ θ X X dθ F B F o n ( ) 1,..., n ( ). n ( θ UCL) Want P LCL 1 α. ˆ ˆ ˆ ˆ ˆ ˆ Note that 1 α P θ α θ θ θ θ α θ ( ) (1 ) 2 2 ˆ ˆ ˆ ˆ ˆ P θ α θ θ θ θ α θ ( ) (1 ) 2 2 = P 2 ˆ θ ˆ θ θ 2 ˆ θ ˆ θ α α (1 ) ( ) 2 2 ( θ UCL) = P LCL. 13

14 An Example by The Simple Method (1) 1 1 X1, X2,..., X101 ~ N( θ, 1), θ= median= F ( ). 2 ˆ 1 1 X(1) X(2)... X(101), θ = Fn ( ) = X(51). 2 Resampling with replacement from,,...,. X X... X. (1) (2) (101) ˆ 1 1 θ = Fn ( ) = X(51). 2 Repeat B = 1000 times, we can get ˆ θ ˆ θ... ˆ θ. (1) (2) (1000) { X X X }

15 An Example by The Simple Method (2) 95% θˆ (1) θˆ (25) { } P ˆ θ ˆ θ ˆ θ 1 α = 95% (25) (975) { ˆ θ ˆ θ ˆ θ ˆ θ ˆ θ ˆ θ} = P (25) (975) θˆ (975) { ˆ θ ˆ ˆ ˆ ˆ} (25) θ θ θ θ(975) θ P { ˆ θ ˆ θ ˆ ˆ } (975) θ θ θ(25) = P 2 2. θˆ (1000) [ LCL = 2 ˆ θ ˆ θ, UCL = 2 ˆ θ ˆ θ ] (975) (25) is an approximate (1- α) confidence interval for θ. 15

16 Flowchart of The Simple Method data x= ( x, x,..., x ) ˆ θ = s( x) 1 2 n resample B times x 1 x 2 x B θ = ˆ get resample statistics b ( b) and then sort them s x θˆ (1) θˆ (2) (2) θˆ ( B ) v = [( B+ 1) α / 2], v = [( B+ 1)(1 α / 2)] (1 α)% confidence interval LCL = 2 ˆ θ ˆ θ, UCL = 2ˆ θ ˆ θ ( v ) ( v )

17 The Simple Method by R 17

18 18

19 The Simple Method by C (1) resample B times: x b ˆ b mean( xb) θ = a= ˆ θ = s( x) = mean( x) 19

20 The Simple Method by C (2) calculate v1, v2 100(1 α)% confidence interval 20

21 21

22 22

23 23

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

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

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. 26

27 The Percentile Method (2) Suppose Y = ˆ θ θ ~ H( i). Then HY ( ) ~ U. Φ HY ( ) ~ Φ ( U)~ N(0, 1). ( ) 1 1 Assume that there exists an unbiased and (monotonly) increasing function g( i) such that g( ˆ θ) g( θ) N(0, 1). 27

28 The Percentile Method (3) If g( ˆ θ) g( θ) N(0, 1), then ( ) ( ) (0, 1). g ˆ θ g ˆ θ N ( ˆ ˆ ) α P g( θ ) g( θ) z 1 α = P ˆ θ g ( g( ˆ θ) + z )) and ˆ ξ = ˆ θ ξα 1 α α ([( B+ 1)(1 α)]) ( ( ˆ θ ) ( θ ) ) α P g g z ( 1 θ ˆ θ ) = P g ( g( ) z )) (Note: z = z for N(0, 1).) α α 1 α 1 = P θ g ( g( ˆ θ) + z ˆ ˆ 1 α)) and ξ1 α = θ ([( B+ 1) α]). ξ1 α 28

29 The Percentile Method (4) ( ˆ ) ([( B+ 1)(1 α )]) Similarly, P θ θ 1 α and P ( ) ([( B+ 1) α/2]) ([( B+ 1)(1 α/2)]) ˆ θ θ ˆ θ 1 α. Summary of the percentile method: P P P ( ) ([( B+ 1) α ]) θ ˆ θ 1 α, ( ) ([( B+ 1)(1 α )]) θ ˆ θ 1 α, ( ) ([( B+ 1) α/2]) ([( B+ 1)(1 α/2)]) ˆ θ θ ˆ θ 1 α. 29

30 Flowchart of The Percentile Method data x= ( x, x,..., x ) ˆ θ = s( x) 1 2 n resample B times x 1 x 2 x B ˆ get resample statistics θ b = s( xb) and then sort them θˆ (1) θˆ (2) (2) θˆ ( B ) v = [( B+ 1) α / 2], v = [( B+ 1)(1 α / 2)] (1 α)% confidence interval LCL = ˆ θ, UCL = ˆ θ ( v ) ( v )

31 The Percentile Method by R 31

32 32

33 The Percentile Method by C resample B times: ˆ b mean( xb) θ = x b calculate v1, v2 100(1 α)% confidence interval 33

34 34

35 35

36 36

37 2.2. The BC Percentile Method Methodology Flowchart R code 37

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. 38

39 Flowchart of The BC Percentile Method data x= ( x, x,..., x ) ˆ θ = s( x) 1 2 n resample B times x 1 x 2 x B θ = ˆ get resample statistics b ( b) and then sort them s x ˆ θˆ θ (1) (2) (2) θˆ ( B ) LCL estimate z 0 v = Φ(2 z z ) α /2 v =Φ(2 z z ) 2 0 α /2 = ˆ θ, UCL = ˆ θ (( B+ 1) v ) (( B+ 1) v ) 1 2 B 1 1 ˆ estimate z ˆ 0 by Φ 1 B b= 1 Φ 1 ( α) = z α { θ } b θ 100(1 α)% confidence interval 39

40 The BC Percentile Method by R 40

41 41

42 2.3. The BCa Percentile Method Methodology Flowchart R code C code 42

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. 43

44 The BCa Percentile Method (2) g( ˆ θ ) g( ˆ θ) P U = + z0 zα 1 α 1 a g( ˆ + θ ) ( ˆ 1 ˆ ˆ ) ( ˆ θ θ θ ) α θ ξα = P g ( g( ) + (1 + a g( ))( z z ) ) = P. 0 g( ˆ θ) g( θ) P U = + z0 zα 1 α 1 + a g( θ ) ˆ 1 g( θ ) ( zα z0) = P θ g ( ) 1 + a ( zα z0) = ˆ + + ˆ z ) ) = P θ ξ. ( ) 0 ( β ) 1 P θ g ( g( θ) (1 a g( θ))( zβ ˆ ξ = ˆ θ. β ([( B+ 1) (1 β )]) 1 1 Similarly, P( θ ˆ θ ) 1 α ([( B+ 1) (1 β )]) and P( ˆ θ θ ˆ θ ) 1 2 α. ([( B+ 1) (1 β )]) ([( B+ 1) (1 β )])

45 The BCa Percentile Method (3) β =? 1 β = 1 PZ ( ) 1 β 1 g( ˆ θ ) ( zα z0) and = g( ˆ θ) + (1 + a g( ˆ θ)( zβ z 1 0)) 1 + a ( z z ) α 0 z z z z zβ = z + and β = 1 P( Z z + ) 1 1 ( ) 1 ( ) α 0 α a zα z0 a zα z0 z z = PZ z + α 0 Similarly, β2 1 ( 0 ). 1 a ( zα z0) 45

46 The BCa Percentile Method (4) z 0 =? ( ) P( ˆ θ ˆ θ) = P g( ˆ θ ) g( ˆ θ) ( P ˆ θ ˆ θ ) 1 0 ˆ ˆ ˆ ˆ g( θ ) g( θ) g( θ) g( θ) = P + z + z 1 a g( ˆ θ) 1 a g( ˆ + + θ) =Φ( z ) z =Φ ( ) and ˆ ˆ B 1 { zˆ } 0 =Φ 1 θb θ. B b= 1 46

47 The BCa Percentile Method (5) a =? n 3 ( ˆ θ ˆ () i θ() i ) i= 1 ˆ Jack =, n ˆ ˆ 2 3/2 6 ( ( θ() i θ() i ) ) i= 1 a where ˆ θ = θ( F ) = θ({ X,..., X and () i n 1, i 1 i n ˆ 1 θ = ˆ θ. () i () i n i= 1,..., X }) n 47

48 Flowchart of The BCa Percentile Method data x= ( x, x,..., x ) ˆ θ = s( x) 1 2 n resample B times x 1 x 2 x B θ = ˆ get resample statistics b ( b) and then sort them s x ˆ θˆ θ (1) (2) (2) θˆ ( B ) estimate z0, a { θb θ} 1 ˆ ˆ B 1 estimate z0 by Φ 1 and a by Jackknife B b= 1 1 ( z z ), 1 ( z z β = Φ z + β = Φ z + ) 1 ( ) 1 ( ) α/2 0 1 α/ azα/2 z0 az1 α /2 z0 Φ 1 ( α) = z α LCL = ˆ θ, UCL = ˆ θ (( B+ 1) (1 β )) (( B+ 1) (1 β )) (1 α)% confidence interval

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

50 50

51 The BCa Percentile Method by R 51

52 52

53 The BCa Percentile Method by C 53

54 54

55 55

56 56

57 57

58 58

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