Nonparametric Inference via Bootstrapping the Debiased Estimator

Transcription

1 Nonparametric Inference via Bootstrapping the Debiased Estimator Yen-Chi Chen Department of Statistics, University of Washington ICSA-Canada Chapter Symposium / 21

2 Problem Setup Let X 1,, X n be an IID random sample from an unknown distribution function with a density function p. For simplicity, we assume p is supported on [0, 1] d. Goal: given a level α, we want to find L α (x), U α (x) using the random sample such that ( P L α (x) p(x) U α (x) x [0, 1] d) 1 α + o(1). Namely, [L α (x), U α (x)] forms an asymptotic simultaneous confidence band of p(x). 2 / 21

3 Simple Approach: using the KDE A classical approach is to construct L α (x), U α (x) using the kernel density estimator (KDE). Let p h (x) = 1 nh d n ( ) Xi x K h i=1 be the KDE where h > 0 is the smoothing bandwidth and K(x) is a smooth function such as a Gaussian. We pick t α such that L α (x) = p h (x) t α, U α (x) = p h (x) + t α. As long as we choose t α wisely, the resulting confidence band is asymptotically valid. 3 / 21

4 Simple Approach: the L Error How do we choose t α to obtain a valid confidence band? A simple idea: inverting the L error. 4 / 21

5 Simple Approach: the L Error How do we choose t α to obtain a valid confidence band? A simple idea: inverting the L error. Let F n (t) be the CDF of p h p = sup x p h (x) p(x). Then the value tα = Fn 1 (1 α) has a nice property: P( p h p t α) = 1 α. 4 / 21

6 Simple Approach: the L Error How do we choose t α to obtain a valid confidence band? A simple idea: inverting the L error. Let F n (t) be the CDF of p h p = sup x p h (x) p(x). Then the value tα = Fn 1 (1 α) has a nice property: This implies Thus, P( p h p t α) = 1 α. P( p h (x) p(x) t α x [0, 1] d ) = 1 α. L α(x) = p h (x) t α, U α(x) = p h (x) + t α leads to a simultaneous confidence band. 4 / 21

7 Simple Approach: the Bootstrap - 1 The previous method is great it works even in a finite sample case. However, it has a critical problem: we do not know the distribution F n! So we cannot compute the quantile. 5 / 21

8 Simple Approach: the Bootstrap - 1 The previous method is great it works even in a finite sample case. However, it has a critical problem: we do not know the distribution F n! So we cannot compute the quantile. A simple solution: using the bootstrap (we will use the empirical bootstrap). 5 / 21

9 Simple Approach: the Bootstrap - 2 Let X1,, X n be a bootstrap sample. We first compute the bootstrap KDE: p h (x) = 1 n ( X nh d K i i=1 x h ). Then we compute the bootstrap L error W = p h p h. After repeating the bootstrap procedure B times, we obtain realizations W 1,, W B. Compute the empirical CDF F n (t) = 1 B B I (W l t). l=1 Finally, we use t α = F 1 n (1 α) and construct the confidence band as L α(x) = p h (x) t α, Ûα(x) = p h (x) + t α. 6 / 21

10 Simple Approach: the Bootstrap - 3 Does the bootstrap approach work? 7 / 21

11 Simple Approach: the Bootstrap - 3 Does the bootstrap approach work? It depends. The bootstrap works if in the sense that p h p h p h p sup P( p h p h < t) P( p h p < t) = o(1). t 7 / 21

12 Simple Approach: the Bootstrap - 3 Does the bootstrap approach work? It depends. The bootstrap works if in the sense that p h p h p h p sup P( p h p h < t) P( p h p < t) = o(1). t However, the above bound holds if we undersmooth the data (Neumann and Polzehl 1998, Chernozhukov et al. 2014). Namely, we choose the smoothing bandwidth h = o(n 1 4+d ). 7 / 21

13 Simple Approach: the Bootstrap - 4 Why do we need to undersmooth the data? 8 / 21

14 Simple Approach: the Bootstrap - 4 Why do we need to undersmooth the data? The L error has a bias-variance tradeoff: ( ) log n p h p = O(h 2 ) +O }{{} P nh d. Bias }{{} stochastic error 8 / 21

15 Simple Approach: the Bootstrap - 4 Why do we need to undersmooth the data? The L error has a bias-variance tradeoff: ( ) log n p h p = O(h 2 ) +O }{{} P nh d. Bias }{{} stochastic error The bootstrap L error is capable of capturing the errors in the stochastic part. However, it does not capture the bias. 8 / 21

16 Simple Approach: the Bootstrap - 4 Why do we need to undersmooth the data? The L error has a bias-variance tradeoff: ( ) log n p h p = O(h 2 ) +O }{{} P nh d. Bias }{{} stochastic error The bootstrap L error is capable of capturing the errors in the stochastic part. However, it does not capture the bias. Undersmooth guarantees that the bias is of a smaller order so we can ignore it. 8 / 21

17 Problem of Undersmoothing ( ) log n p h p = O(h 2 ) + O }{{} P nh d. Bias }{{} stochastic error Undermoothing has a problem: we do not have the optimal convergence rate. The optimal rate occurs when we balance the bias and stochastic error: h = h opt n 1 d+4 (ignoring the log n factor). 9 / 21

18 Problem of Undersmoothing ( ) log n p h p = O(h 2 ) + O }{{} P nh d. Bias }{{} stochastic error Undermoothing has a problem: we do not have the optimal convergence rate. The optimal rate occurs when we balance the bias and stochastic error: h = h opt n 1 d+4 (ignoring the log n factor). A remedy to this problem: choose h optimally but correct the bias (debiased method). 9 / 21

19 The Debiased Method - 1 The idea of the debiased method is based on the fact that a leading term of O(h 2 ) is h 2 2 C K 2 p(x), where C K is a known constant depending on the kernel function and 2 is the Laplacian operator. 10 / 21

20 The Debiased Method - 1 The idea of the debiased method is based on the fact that a leading term of O(h 2 ) is h 2 2 C K 2 p(x), where C K is a known constant depending on the kernel function and 2 is the Laplacian operator. We can estimate 2 p via applying the Laplacian operator to a KDE p h. However, such an estimator is inconsistent when we choose h opt n 1 d+4 because 2 p h (x) 2 p(x) = O(h 2 ) + O P ( 1 nh d+4 ). 10 / 21

21 The Debiased Method - 1 The idea of the debiased method is based on the fact that a leading term of O(h 2 ) is h 2 2 C K 2 p(x), where C K is a known constant depending on the kernel function and 2 is the Laplacian operator. We can estimate 2 p via applying the Laplacian operator to a KDE p h. However, such an estimator is inconsistent when we choose h opt n 1 d+4 because 2 p h (x) 2 p(x) = O(h 2 ) + O P ( 1 nh d+4 The choice h = h opt n 1 d+4 implies 2 p h (x) 2 p(x) = o(1) + O P (1). ). 10 / 21

22 The Debiased Method - 2 To handle this situation, people suggested using two KDE s, one for estimating the density and the other for estimating the bias. 1 This idea has been used in Calonico et al. (2015) for a pointwise confidence interval. 11 / 21

23 The Debiased Method - 2 To handle this situation, people suggested using two KDE s, one for estimating the density and the other for estimating the bias. However, actually we ONLY need one KDE. 1 This idea has been used in Calonico et al. (2015) for a pointwise confidence interval. 11 / 21

24 The Debiased Method - 2 To handle this situation, people suggested using two KDE s, one for estimating the density and the other for estimating the bias. However, actually we ONLY need one KDE. We propose using the same KDE p h (x) to debias the estimator 1. 1 This idea has been used in Calonico et al. (2015) for a pointwise confidence interval. 11 / 21

25 The Debiased Method - 2 To handle this situation, people suggested using two KDE s, one for estimating the density and the other for estimating the bias. However, actually we ONLY need one KDE. We propose using the same KDE p h (x) to debias the estimator 1. Namely, we propose to use with h = h opt n 1 d+4. p h (x) = p h (x) h2 2 C K 2 p h (x) The estimator p h (x) is called the debiased estimator. 1 This idea has been used in Calonico et al. (2015) for a pointwise confidence interval. 11 / 21

26 The Debiased Method + Bootstrap To construct a confidence band, we use the bootstrap again but this time we compute the bootstrap debiased estimator and evaluate p h p h. p h (x) = p h2 h (x) 2 C K 2 p h (x) After repeating the bootstrap procedure many times, we compute the EDF F n of the realizations of p h p h and obtain the quantile t α = (1 α). The confidence band is F 1 n L α (x) = p h (x) t α, Ũ α (x) = p h (x) + t α. 12 / 21

27 Theory of the Debiased Method Theorem (Chen 2017) Assume p belongs to β-hölder class with β > 2 and the kernel function satisfies smoothness conditions. When h n 1 d+4, P ( Lα (x) p(x) Ũα(x) x [0, 1] d) = 1 α + o(1). Namely, the debiased estimator leads to an asymptotic simultaneous confidence band under the choice h n 1 d / 21

28 Why the Debiased Method Work? - 1 Why the debiased method work? Didn t we have an inconsistent bias estimator? 14 / 21

29 Why the Debiased Method Work? - 1 Why the debiased method work? Didn t we have an inconsistent bias estimator? We indeed do not have a consistent bias estimator but this is fine! 14 / 21

30 Why the Debiased Method Work? - 1 Why the debiased method work? Didn t we have an inconsistent bias estimator? We indeed do not have a consistent bias estimator but this is fine! Recall that when h n 1 d+4, 2 p h (x) 2 p(x) = o(1) + }{{} O P (1) }{{}. bias stochastic variation 14 / 21

31 Why the Debiased Method Work? - 1 Why the debiased method work? Didn t we have an inconsistent bias estimator? We indeed do not have a consistent bias estimator but this is fine! Recall that when h n 1 d+4, 2 p h (x) 2 p(x) = o(1) + }{{} O P (1) }{{}. bias stochastic variation Thus, our debiased estimator has three errors: p h (x) p(x) = p h (x) h2 2 C K p h (x) p(x) = h2 ( 1 2 C K 2 p(x) + o(h 2 ) +O P }{{} nh d bias ) h2 2 C K 2 p h (x) 14 / 21

32 Why the Debiased Method Work? - 2 The above equation equals (h n 1 d+4 ) ( ) p h (x) p(x) = h2 1 2 C K p(x) + o(h 2 ) +O P }{{} nh d h2 2 C K p h (x) bias = o(h 2 ) + O P ( 1 nh d ) + h2 2 C ( K 2 p(x) 2 p h (x) ) }{{} =o(1)+o P (1) ( ) 1 = o(h 2 ) + O P nh d + o(h 2 ) + O P (h 2 ) ( ) 1 = o(h 2 ( ) + O P nh d + O P h 2 ). Both the orange and purple terms are stochastic variation. Orange: from estimating the density. Purple: from estimating the bias. 15 / 21

33 Why the Debiased Method Work? - 3 When h n 1 d+4, the error rate ( ) 1 p h (x) p(x) = o(h 2 ( ) + O P nh d + O P h 2 ) = O P (n 2 d+4 ) is dominated by the stochastic variation. As a result, the bootstrap can capture the errors, leading to an asymptotic valid confidence band. 16 / 21

34 Why the Debiased Method Work? - 4 Actually, after closely inspecting the debiased estimator, you can find that p h (x) = p h (x) h2 2 C K 2 p h (x) = 1 n ( ) Xi x nh d K h2 h 2 C 1 K nh d i=1 = 1 n ( ) Xi x nh d M, h i=1 n ( ) 2 Xi x K h where M(x) = K(x) C K 2 2 K(x). Namely, the debiased estimator is a KDE with kernel function M(x)! i=1 17 / 21

35 Why the Debiased Method Work? - 5 The kernel function M(x) = K(x) C K 2 2 K(x) is actually a higher order kernel. 18 / 21

36 Why the Debiased Method Work? - 5 The kernel function M(x) = K(x) C K 2 2 K(x) is actually a higher order kernel. You can show that if the kernel function K(x) is a γ-th order kernel function, then the corresponding M(x) will be a (γ + 2)-th order kernel (Calonico et al. 2015, Scott 2015). 18 / 21

37 Why the Debiased Method Work? - 5 The kernel function M(x) = K(x) C K 2 2 K(x) is actually a higher order kernel. You can show that if the kernel function K(x) is a γ-th order kernel function, then the corresponding M(x) will be a (γ + 2)-th order kernel (Calonico et al. 2015, Scott 2015). Because the debiased estimator p h (x) uses a higher order kernel, the bias is moved to the next order, leaving the stochastic variation dominating the error. 18 / 21

38 Simulation Density Estimation, 95% CI Confidence Sets, Gaussian, N= Original KDE Debiased KDE Bootstrap Coverage Original KDE Debiased KDE X Nominal Coverage 19 / 21

39 Conclusion We illustrate a bootstrap approach to construct a simultaneous confidence band via a debiased KDE. This approach allows us to choose the smoothing bandwidth optimally and still leads to an asymptotic confidence band. A similar idea can also be applied to regression problem and local polynomial estimator. More details can be found in Chen, Yen-Chi. "Nonparametric Inference via Bootstrapping the Debiased Estimator." arxiv preprint arxiv: (2017). 20 / 21

40 Thank you! 21 / 21

41 References 1 Y.-C. Chen. Nonparametric Inference via Bootstrapping the Debiased Estimator. arxiv preprint arxiv: , Y.-C. Chen. A Tutorial on Kernel Density Estimation and Recent Advances. arxiv preprint arxiv: , S. Calonico, M. D. Cattaneo, and M. H. Farrell. On the effect of bias estimation on coverage accuracy in nonparametric inference. arxiv preprint arxiv: , V. Chernozhukov, D. Chetverikov, and K. Kato. Anti-concentration and honest, adaptive confidence bands. The Annals of Statistics, 42(5): , M. H. Neumann and J. Polzehl. Simultaneous bootstrap confidence bands in nonparametric regression. Journal of Nonparametric Statistics, 9(4): , D. W. Scott. Multivariate density estimation: theory, practice, and visualization. John Wiley & Sons, / 21