The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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An Adaptive Bandwidth Selection Algorithm in Nonparametric Regression

비모수적 회귀선의 추정을 위한 bandwidth 선택 알고리즘

  • Kyung Joon Cha (133-791 Department of Mathematics, Hanyang University, Seongdong-ku, Haengdang-dong, 17, Seoul, Korea) ;
  • Seung Woo Lee (133-791 Department of Mathematics, Hanyang University, Seongdong-ku, Haengdang-dong, 17, Seoul, Korea)
  • Published : 1994.02.01
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Abstract

Nonparametric regression technique using kernel estimator is an attractive alternative that has received some attention, recently. The kernel estimate depends on two quantities which have to be provided by the user : the kernel function and the bandwidth. However, the more difficult problem is how to find an appropriate bandwidth which controls the amount of smoothing (see Silverman, 1986). Thus, in practical situation, it is certainly desirable to determine an appropriate bandwidth in some automatic fashion. Thus, the problem is to find a data-driven or adaptive (i.e., depending only on the data and then directly computable in practice) bandwidth that performs reasonably well relative to the best theoretical bandwidth. In this paper, we introduce a relation between bias and variance of mean square error. Thus, we present a simple and effective algorithm for selecting local bandwidths in kernel regression.

커널 추정은 커널함수와 bandwidth에 의해서 결정이 된다. 그러나 평활의 정도를 조절하는 적절한 bandwidth를 찾는 것이 더욱 중요한 문제이다. 그러므로 이론적으로 최적의 bandwidth와 비교하여 실제자료에 잘 적용될 수 있는 적절한 bandwidth를 어떻게 찾느냐는 것이 문제가 된다. 본 논문에서는 평균제곱오차(mean square error)의 편의(bias)와 분산(variance)의 관계를 통하여 커널을 이용한 회귀선의 추정에 있어서 간단하고 효과적인 local bandwidth를 찾을 수 있는 알고리즘을 제안하였다.

Keywords

References

  1. Spline Smoothing and Nonparametric Regression Eubank,R.L.
  2. Kernel Estimation of Regression Function: In Smoothing Techniques for Curve Estimation Gasser,Th.;M$2\"{u}$ller,H.G.
  3. Journal of the Royal Statistical Society v.B47 Kernels for Nonparametric Curve Estimation Gasser,Th.;M$2\"{u}$ller,H.G.;Mammitzsch,V.
  4. Biometrika v.73 Residual variance and Residual Pattern in Nonlinear Regression Gasser,Th.;Sroka,L.;Jennen-Steinmetz,C.
  5. Statistics and Probability Letters v.8 A Local Cross-Validation Algorithm Hall,P.;Schucany,W.R.
  6. Journal of the American Statistical Association v.83 Bootstraping in Nonparametric Regression:Local Adaptive Smoothing and Confidence Bands Hardle,W.;Bowman,A.W.
  7. A Survey, Empirical Economics v.13 Automatic Smoothing parameter Selection Marron,J.S.
  8. Statistics and Decisions v.2 Empirical Bandwidth Choice for Nonparametric Kernel Regression by Means of Pilot Estimators Muller,H.G.
  9. Nonparametric Regression Analysis of Longitudinal Data Muller,H.G.
  10. Journal of the Royal Statistical Society v.B34 Nonprametric Function Fitting Priestly,M.B.;Chao,M.T.
  11. Density Estimation for Statistics and Data Analysis Silverman,B.W.
  12. Journal of the American Statistical Association v.84 Local Bandwidth Selection for a Kernel Estimates Staniswalis,J.G.