The Annals of Statistics

On automatic boundary corrections

Ming-Yen Cheng, Jianqing Fan, and J. S. Marron

Full-text: Open access

Abstract

Many popular curve estimators based on smoothing have difficulties caused by boundary effects. These effects are visually disturbing in practice and can play a dominant role in theoretical analysis. Local polynomial regression smoothers are known to correct boundary effects automatically. Some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections. In both settings of density and regression estimation, we investigate best weight functions for local polynomial fitting at the endpoints and find a simple solution. The solution is universal for general degree of local polynomial fitting and general order of estimated derivative. Furthermore, such local polynomial estimators are best among all linear estimators in a weak minimax sense, and they are highly efficient even in the usual linear minimax sense.

Article information

Source
Ann. Statist., Volume 25, Number 4 (1997), 1691-1708.

Dates
First available in Project Euclid: 9 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1031594737

Digital Object Identifier
doi:10.1214/aos/1031594737

Mathematical Reviews number (MathSciNet)
MR1463570

Zentralblatt MATH identifier
0890.62026

Subjects
Primary: 62G07: Density estimation
Secondary: 62C20: Minimax procedures

Keywords
Boundary correction data binning local polynomial fit minimax risk weak minimaxity

Citation

Cheng, Ming-Yen; Fan, Jianqing; Marron, J. S. On automatic boundary corrections. Ann. Statist. 25 (1997), no. 4, 1691--1708. doi:10.1214/aos/1031594737. https://projecteuclid.org/euclid.aos/1031594737


Export citation

References

  • Brown, L. D. and Low, M. G. (1996). Asy mptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398.
  • Brown, L., Low, M. and Zhao, L. (1997). Anomalous behavior of fixed parameter asy mptotic rates of convergence in nonparametric function estimation problems. Unpublished manuscript.
  • Cheng, M. Y. (1994). On boundary effects of smooth curve estimators. Ph.D. dissertation, Dept. Statistics, Univ. North Carolina, Chapel Hill.
  • Cheng, M. Y. (1997). Boundary-aware estimators of integrated density derivative products. J. Roy. Statist. Soc. Ser. B 59 191-203.
  • Cheng, M. Y., Fan, J. and Marron, J. S. (1993). Minimax efficiency of local poly nomial fit estimators at boundaries. Mimeo Series 2098, Inst. Statist., Univ. North Carolina, Chapel Hill.
  • Chu, C. K. and Marron, J. S. (1991). Choosing a kernel regression estimator. Statist. Sci. 6 404-436.
  • Djojosugito, R. A. and Speckman, P. L. (1992). Boundary bias correction in nonparametric density estimation. Comm. Statist. Theory Methods 21 69-88.
  • Donoho, D. L. and Johnstone, I. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455.
  • Donoho, D. L. and Liu, R. C. (1991). Geometrizing rate of convergence III. Ann. Statist. 19 668-701.
  • Donoho, D. L., Liu, R. C. and MacGibbon, B. (1990). Minimax risk over hy per rectangles, implications. Ann. Statist. 18 1416-1437.
  • Efroimovich, S. (1996). On nonparametric regression for i.i.d. observations in general setting. Ann. Statist. 24 1126-1144.
  • Epanechnikov, V. A. (1969). Nonparametric estimation of a multidimensional probability density. Theory Probab. Appl. 14 153-158.
  • Eubank, R. L. and Speckman, P. (1991). A bias reduction theorem with applications in nonparametric regression. Scand. J. Statist. 18 211-222.
  • Fan, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87 998-1004.
  • Fan, J. (1993). Local linear regression smoothers and their minimax efficiency. Ann. Statist. 21 196-216.
  • Fan, J. and Gijbels, I. (1992). Variable bandwidth and local linear regression smoothers. Ann. Statist. 20 2008-2036.
  • Fan, J. and Hall, P. (1994). On curve estimation by minimizing mean absolute deviation and its implications. Ann. Statist. 22 867-885.
  • Fan, J. and Marron, J. S. (1994). Fast implementations of nonparametric curve estimators. J. Comput. Graph. Statist. 3 35-56.
  • Fan, J., Gasser, T., Gijbels, I., Brockmann, M. and Engels, J. (1997). Local poly nomial fitting: a standard for nonparametric regression. Ann. Inst. Statist. Math. 49 79-99.
  • Gasser, T. and M ¨uller, H. G. (1979). Kernel estimation of regression functions. Smoothing Techniques for Curve Estimation. Lecture Notes in Math. 757 23-68. Springer, New York.
  • Gasser, T., M ¨uller, H. G. and Mammitzsch, V. (1985). Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B 47 238-252.
  • Granovsky, B. L. and M ¨uller, H. G. (1991). Optimizing kernel methods: a unifying variational principle. Internat. Statist. Rev. 59 373-388.
  • Hall, P. and Wehrly, T. E. (1991). A geometrical method for removing edge effects from kernelty pe nonparametric regression estimators. J. Amer. Statist. Assoc. 86 665-672.
  • Hastie, T. and Loader, C. (1993). Local regression: automatic kernel carpentry. Statist. Sci. 8 120-143.
  • Jones, M. C. (1993). Simple boundary correction for kernel density estimation. Statist. Comput. 3 135-146.
  • Lejeune, M. and Sarda, P. (1992). Smooth estimators of distribution and density functions. Comput. Statist. Data Anal. 14 457-471.
  • M ¨uller, H. G. (1991). Smooth optimal kernel estimators near endpoints. Biometrika 78 521-530.
  • Nussbaum, M. (1985). Spline smoothing in regression models and asy mptotic efficiency in L2 Ann. Statist. 13 984-997.
  • Rice, J. (1984). Boundary modification for kernel regression. Comm. Statist. Theory Methods 13 893-900.
  • Rice, J. and Rosenblatt, M. (1981). Integrated mean squared error of a smoothing spline. J. Approx. Theory 33 353-365.
  • Ruppert, D. and Wand, M. P. (1994). Multivariate weighted least squares regression. Ann. Statist. 22 1346-1370.
  • Sacks, J. and Ylvisaker, D. (1981). Asy mptotically optimum kernels for density estimation at a point. Ann. Statist. 9 334-346.
  • Schuster, E. F. (1985). Incorporating support constraints into nonparametric estimators of densities. Comm. Statist. Theory Methods 14 1123-1136.
  • Stone, C. J. (1977). Consistent nonparametric regression. Ann. Statist. 5 595-645.