The Annals of Statistics

New methods for bias correction at endpoints and boundaries

Peter Hall and Byeong U. Park

Full-text: Open access


We suggest two new, translation-based methods for estimating and correcting for bias when estimating the edge of a distribution. The first uses an empirical translation applied to the argument of the kernel, in order to remove the main effects of the asymmetries that are inherent when constructing estimators at boundaries. Placing the translation inside the kernel is in marked contrast to traditional approaches, such as the use of high-order kernels, which are related to the jackknife and, in effect, apply the translation outside the kernel. Our approach has the advantage of producing bias estimators that, while enjoying a high order of accuracy, are guaranteed to respect the sign of bias. Our second method is a new bootstrap technique. It involves translating an initial boundary estimate toward the body of the dataset, constructing repeated boundary estimates from data that lie below the respective translations, and employing averages of the resulting empirical bias approximations to estimate the bias of the original estimator. The first of the two methods is most appropriate in univariate cases, and is studied there; the second approach may be used to bias-correct estimates of boundaries of multivariate distributions, and is explored in the bivariate case.

Article information

Ann. Statist., Volume 30, Number 5 (2002), 1460-1479.

First available in Project Euclid: 28 October 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Bias estimation bootstrap curve estimation free disposal hull estimator frontier estimation kernel methods nonparametric density estimation productivity analysis translation


Hall, Peter; Park, Byeong U. New methods for bias correction at endpoints and boundaries. Ann. Statist. 30 (2002), no. 5, 1460--1479. doi:10.1214/aos/1035844983.

Export citation


  • ATHREy A, K. B. (1987a). Bootstrap of the mean in the infinite variance case. Ann. Statist. 15 724- 731.
  • ATHREy A, K. B. (1987b). Bootstrap of the mean in the infinite variance case. In Proceedings of the First World Congress of Bernoulli Society 2 95-98. VNU Scientific Press, Utrecht.
  • BICKEL, P., GÖTZE, F. and VAN ZWET, W. R. (1997). Resampling fewer than n observations: Gains, losses, and remedies for losses. Statist. Sinica 7 1-31.
  • BLOCH, D. A. and GASTWIRTH, J. L. (1968). On a simple estimate of the reciprocal of the density function. Ann. Math. Statist. 39 1083-1085.
  • BOFINGER, E. (1975). Estimation of a density function using order statistics. Austral. J. Statist. 17 1-7.
  • CHENG, C. (1995). Uniform consistency of generalized kernel estimators of quantile density. Ann. Statist. 23 2285-2291.
  • CHENG, C. and PARZEN, E. (1997). Unified estimators of smooth quantile and quantile density functions. J. Statist. Plann. Inference 59 291-307.
  • CHEVALIER, J. (1976). Estimation du support et du contenu du support d'une loi de probabilité. Ann. Inst. H. Poincaré Probab. Statist. 12 339-364.
  • CHOI, E. and HALL, P. (1999). Data sharpening as a prelude to density estimation. Biometrika 86 941-947.
  • CSÖRG O, M. (1983). Quantile Processes with Statistical Applications. SIAM, Philadelphia.
  • DEPRINS, D., SIMAR, L. and TULKENS, H. (1984). Measuring labor efficiency in post offices. In The Performance of Public Enterprises: Concepts and Measurements (M. Marchand, P. Pestieau and H. Tulkens, eds.) 243-267. North-Holland, Amsterdam.
  • FALK, M. (1986). On the estimation of the quantile density function. Statist. Probab. Lett. 4 69-73.
  • GIJBELS, I., MAMMEN, E., PARK, B. U. and SIMAR, L. (1999). On estimation of monotone and concave frontier functions. J. Amer. Statist. Assoc. 94 220-228.
  • HALL, P. (1989). On convergence rates in nonparametric problems. Internat. Statist. Rev. 57 45-58.
  • HALL, P. (1990). Asy mptotic properties of the bootstrap for heavy-tailed distributions. Ann. Probab. 18 1342-1360.
  • HALL, P., PARK, B. U. and STERN, S. E. (1998). On poly nomial estimators of frontiers and boundaries. J. Multivariate Anal. 66 71-98.
  • HALL, P., PARK, B. U. and TURLACH, B. A. (1998). Rolling-ball method for estimating the boundary of the support of a point-process intensity. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • HÄRDLE, W., PARK, B. U. and TSy BAKOV, A. B. (1995). Estimation of nonsharp support boundaries. J. Multivariate Anal. 55 205-218.
  • JONES, M. C. (1992). Estimating densities, quantiles, quantile densities and density quantiles. Ann. Inst. Statist. Math. 44 721-727.
  • JONES, M. C. and SIGNORINI, D. F. (1997). A comparison of higher-order bias kernel density estimators. J. Amer. Statist. Assoc. 92 1063-1073.
  • KNEIP, A., PARK, B. U. and SIMAR, L. (1998). A note on the convergence of nonparametric DEA estimators for production efficiency scores. Econom. Theory 14 783-793.
  • KNIGHT, K. (1989). On the bootstrap of the sample mean in the infinite variance case. Ann. Statist. 17 1168-1175.
  • KOROSTELEV, A. P., SIMAR, L. and TSy BAKOV, A. B. (1995). Efficient estimation of monotone boundaries. Ann. Statist. 23 476-489.
  • KOROSTELEV, A. P. and TSy BAKOV, A. B. (1993). Minimax Theory of Image Reconstruction. Springer, New York.
  • LINTON, O. and NIELSEN, J. P. (1994). A multiplicative bias reduction method for nonparametric regression. Statist. Probab. Lett. 19 181-187.
  • MAMMEN, E. and TSy BAKOV, A. B. (1995). Asy mptotically minimax recovery of sets with smooth boundaries. Ann. Statist. 23 502-524.
  • PARK, B. U., SIMAR, L. and WEINER, C. (2000). The FDH estimator for productivity efficiency scores: Asy mptotic properties. Econom. Theory 16 855-877.
  • POLITIS, D., ROMANO, J. P. and WOLF, M. (1999). Subsampling. Springer, New York.
  • REISS, R.-D. (1978). Approximate distribution of the maximum deviation of histograms. Metrika 25 9-26.
  • RIPLEY, B. D. and RASSON, J. P. (1977). Finding the edge of a Poisson forest. J. Appl. Probab. 14 483-491.
  • SAMIUDDIN, M. and EL-SAy YAD, G. M. (1990). On nonparametric kernel density estimates. Biometrika 77 865-874.
  • SIDDIQUI, M. M. (1960). Distribution of quantiles in samples from a bivariate population. Journal of Research of the National Bureau Standards 64B 145-150.
  • SIMAR, L. and WILSON, P. W. (1998). A general methodology for bootstrapping in nonparametric frontier models. Research Report DP 9811, Institut de Statistique, Univ. Catholique de Louvain, Belgium.
  • WAND, M. P. and JONES, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.
  • WELSH, A. H. (1988). Asy mptotically efficient estimation of the sparsity function at a point. Statist. Sinica 6 427-432.