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Simple boundary correction for kernel density estimation

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Abstract

If a probability density function has bounded support, kernel density estimates often overspill the boundaries and are consequently especially biased at and near these edges. In this paper, we consider the alleviation of this boundary problem. A simple unified framework is provided which covers a number of straightforward methods and allows for their comparison: ‘generalized jackknifing’ generates a variety of simple boundary kernel formulae. A well-known method of Rice (1984) is a special case. A popular linear correction method is another: it has close connections with the boundary properties of local linear fitting (Fan and Gijbels, 1992). Links with the ‘optimal’ boundary kernels of Müller (1991) are investigated. Novel boundary kernels involving kernel derivatives and generalized reflection arise too. In comparisons, various generalized jackknifing methods perform rather similarly, so this, together with its existing popularity, make linear correction as good a method as any. In an as yet unsuccessful attempt to improve on generalized jackknifing, a variety of alternative approaches is considered. A further contribution is to consider generalized jackknife boundary correction for density derivative estimation. En route to all this, a natural analogue of local polynomial regression for density estimation is defined and discussed.

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References

  • Azari, A. S., Mack, Y. P. and Müller, H.-G. (1992) Ultraspherical polynomial, kernel and hybrid estimators for nonparametric regression. Sankhyā A, 54, 80–96.

    Google Scholar 

  • Boneva, L. I., Kendall, D. G. and Stefanov, I. (1971) Spline transformations: three new diagnostic aids for the statistical dataanalyst (with discussion). Journal of the Royal Statistical Society B, 33, 1–70.

    Google Scholar 

  • Cleveland, W. S. (1979) Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association, 74, 829–836.

    Google Scholar 

  • Cline, D. B. and Hart, J. D. (1991) Kernel estimation of densities with discontinuities or discontinuous derivatives. Statistics, 22, 69–84.

    Google Scholar 

  • Copas, J. B. and Fryer, M. J. (1980) Density estimation and suicide risks in psychiatric treatment. Journal of the Royal Statistical Society, A, 143, 167–176.

    Google Scholar 

  • Diggle, P. J. (1985) A kernel method for smoothing point process data. Applied Statistics, 34, 138–147.

    Google Scholar 

  • Diggle, P. J. and Marron, J. S. (1988) Equivalence of smoothing parameter selectors in density and intensity estimation. Journal of the American Statistical Association, 83, 793–800.

    Google Scholar 

  • Eubank, R. L. and Speckman, P. (1991) A bias reduction theorem with applications in nonparametric regression. Scandinavian Journal of Statistics, 18, 211–222.

    Google Scholar 

  • Fan, J. (1992) Design-adaptive nonparametric regression. Journal of the American Statistical Association, 87, 998–1004.

    Google Scholar 

  • Fan, J. (1993) A remedy to regression estimators and nonparametric minimax efficiency. Annals of Statistics, 21, 196–216.

    Google Scholar 

  • Fan, J. and Gijbels, I. (1992) Variable bandwidth and local linear regression smoothers. Annals of Statistics, 20, 2008–2036.

    Google Scholar 

  • Gasser, T. and Müller, H.-G. (1979) Kernel estimation of regression functions. In Smoothing Techniques for Curve Estimation, T. Gasser and M. Rosenblatt (eds), Springer, Berlin, pp. 23–68.

    Google Scholar 

  • Gasser, T., Müller, H.-G. and Mammitzsch, V. (1985) Kernels for nonparametric curve estimation. Journal of the Royal Statistical Society B, 47, 238–252.

    Google Scholar 

  • Ghosh, B. K. and Huang, W.-M. (1992) Optimum bandwidths and kernels for estimating certain discontinuous densities. Annals of the Institute of Statistical Mathematics, 44, 563–577.

    Google Scholar 

  • Granovsky, B. L. and Müller, H.-G. (1991) Optimizing kernel methods: a unifying variational principle. International Statistical Review, 59, 373–388.

    Google Scholar 

  • Hall, P. and Wehrly, T. E. (1991) A geometrical method for removing edge effects from kernel-type nonparametric regression estimators. Journal of the American Statistical Association, 86, 665–672.

    Google Scholar 

  • Härdle, W. (1990) Applied Nonparametric Regression. Cambridge University Press.

  • Hart, J. D. and Wehrly, T. E. (1992) Kernel regression estimation when the boundary is large, with an application to testing the adequacy of polynomial models. Journal of the American Statistical Association, 87, 1018–1024.

    Google Scholar 

  • Hastie, T. J. and Loader, C. (1993) Local regression: automatic kernel carpentry (with comments). Statistical Science, 8, 120–143.

    Google Scholar 

  • Hominal, P. and Deheuvels, P. (1979) Estimation non paramétrique de la densité compte-tenu d'informations sur la support. Revue Statistique Appliqué, 27, 47–68.

    Google Scholar 

  • Jones, M. C. (1990) Variable kernel density estimates and variable kernel density estimates. Australian Journal of Statistics, 32, 361–371. Correction, 33, 119.

    Google Scholar 

  • Jones, M. C. (1991) Kernel density estimation for length biased data. Biometrika, 78, 511–519.

    Google Scholar 

  • Jones, M. C. and Foster, P. J. (1993) Generalized jackknifing and higher order kernels. Journal of Nonparametric Statistics, to appear.

  • Lejeune, M. (1985) Estimation non-paramétrique par noyaux: regression polynomial mobile. Revue Statistique Appliqué, 33, 43–66.

    Google Scholar 

  • Lejeune, M. and Sarda, P. (1992) Smooth estimators of distribution and density functions. Computational Statistics and Data Analysis, 14, 457–471.

    Google Scholar 

  • Marron, J. S. and Nolan, D. (1989) Canonical kernels for density estimation. Statistics and Probability Letters, 7, 195–199.

    Google Scholar 

  • Marron, J. S. and Ruppert, D. (1992) Transformations to reduce boundary bias in kernel density estimation, to appear.

  • Marron, J. S. and Wand, M. P. (1992) Exact mean integrated squared error. Annals of Statistics, 20, 712–736.

    Google Scholar 

  • Müller, H.-G. (1987) Weighted local regression and kernel methods for nonparametric curve fitting. Journal of the American Statistical Association, 82, 231–238.

    Google Scholar 

  • Müller, H.-G. (1991) Smooth optimum kernel estimators near endpoints. Biometrika, 78, 521–530.

    Google Scholar 

  • Müller, H.-G. (1993) On the boundary kernel method for non-parametric curve estimation near endpoints. Scandinavian Journal of Statistics, to appear.

  • Muttlak, H. A. and McDonald, L. L. (1990) Ranked set sampling with size-biased probability of selection. Biometrics, 46, 435–445.

    Google Scholar 

  • Rice, J. A. (1984) Boundary modification for kernel regression. Communications in Statistics — Theory and Methods, 13, 893–900.

    Google Scholar 

  • Ruppert, D. and Cline, D. H. (1992) Transformation-kernel density estimation — bias reduction by empirical transformations. Annals of Statistics, to appear.

  • Ruppert, D. and Wand, M. P. (1993) Multivariate locally weighted least squares regression. Annals of Statistics, to appear.

  • Sarda, P. (1991) Estimating smooth distribution functions, in Nonparametric Functional Estimation and Related Topics, G. G. Roussas (ed.), Kluwer, Dordrecht, pp. 261–270.

    Google Scholar 

  • Schucany, W. R. (1989) On nonparametric regression with higherorder kernels. Journal of Statistical Planning and Inference, 23, 145–151.

    Google Scholar 

  • Schucany, W. R. and Sommers, J. P. (1977) Improvement of kernel type estimators. Journal of the American Statistical Association, 72, 420–423.

    Google Scholar 

  • Schucany, W. R., Gray, H. L. and Owen, D. B. (1971) On bias reduction in estimation. Journal of the American Statistical Association, 66, 524–533.

    Google Scholar 

  • Schuster, E. F. (1985) Incorporating support constraints into non-parametric estimators of densities. Communications in Statistics — Theory and Methods, 14, 1123–1136.

    Google Scholar 

  • Scott, D. W. (1992) Multivariate Density Estimation; Theory, Practice and Visualization. Wiley, New York.

    Google Scholar 

  • Sheather, S. J. and Jones, M. C. (1991) A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society B, 53, 683–690.

    Google Scholar 

  • Silverman, B. W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.

    Google Scholar 

  • Wand, M. P., Marron, J. S. and Ruppert, D. (1991) Transformations in density estimation (with comments). Journal of the American Statistical Association, 86, 343–361.

    Google Scholar 

Download references

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Jones, M.C. Simple boundary correction for kernel density estimation. Stat Comput 3, 135–146 (1993). https://doi.org/10.1007/BF00147776

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