• Bernoulli
  • Volume 15, Number 3 (2009), 614-633.

Nonparametric “regression” when errors are positioned at end-points

Peter Hall and Ingrid Van Keilegom

Full-text: Open access


Increasing practical interest has been shown in regression problems where the errors, or disturbances, are centred in a way that reflects particular characteristics of the mechanism that generated the data. In economics this occurs in problems involving data on markets, productivity and auctions, where it can be natural to centre at an end-point of the error distribution rather than at the distribution’s mean. Often these cases have an extreme-value character, and in that broader context, examples involving meteorological, record-value and production-frontier data have been discussed in the literature. We shall discuss nonparametric methods for estimating regression curves in these settings, showing that they have features that contrast so starkly with those in better understood problems that they lead to apparent contradictions. For example, merely by centring errors at their end-points rather than their means the problem can change from one with a familiar nonparametric character, where the optimal convergence rate is slower than $n^{−1/2}$, to one in the super-efficient class, where the optimal rate is faster than $n^{−1/2}$. Moreover, when the errors are centred in a non-standard way there is greater intrinsic interest in estimating characteristics of the error distribution, as well as of the regression mean itself. The paper will also address this aspect of the problem.

Article information

Bernoulli, Volume 15, Number 3 (2009), 614-633.

First available in Project Euclid: 28 August 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

bandwidth curve estimation extreme-value theory jump discontinuity kernel local linear methods local polynomial methods nonparametric regression smoothing super efficiency


Hall, Peter; Van Keilegom, Ingrid. Nonparametric “regression” when errors are positioned at end-points. Bernoulli 15 (2009), no. 3, 614--633. doi:10.3150/08-BEJ173.

Export citation


  • Aigner, D., Lovell, C.A., Knox, K. and Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. J. Econometrics 6 21–37.
  • Breen, R. (1996). Regression Models: Censored, Sample Selected, or Truncated Data. Thousand Oaks, CA: Sage Publications.
  • Campo, S., Guerre, E., Perrigne, I. and Vuong, Q. (2002). Semiparametric estimation of first-price auctions with risk averse bidders. Available at
  • Chernozhukov, V. (1998). Nonparametric extreme regression quantiles. Technical report, Stanford Univ.
  • Chernozhukov, V. (2005). Extremal quantile regression. Ann. Statist. 33 806–839.
  • Chernozhukov, V. and Hong, A. (2004). Likelihood estimation and inference in a class of nonregular econometric models. Econometrica 72 1445–1480.
  • Christensen, L. and Greene, R. (1976). Economies of scale in U.S. electric power generation. J. Polit. Econom. 84 653–667.
  • Donald, S.G. and Paarsch, H.J. (1993). Piecewise pseudo-maximum likelihood estimation in empirical models of auctions. Internat. Econom. Rev. 34 121–148.
  • Donald, S.G. and Paarsch, H.J. (2002). Superconsistent estimation and inference in structural econometric models using extreme order statistics. J. Econometrics 109 305–340.
  • 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.
  • Gijbels, I. and Peng, L. (2000). Estimation of a support curve via order statistics. Extremes 3 251–277.
  • Greene, W.H. (1990). A Gamma-distributed stochastic frontier model. J. Econometrics 46 141–163.
  • Hall, P. (1978). Representations and limit theorems for extreme value distributions. J. Appl. Probab. 15 639–644.
  • Hall, P. (1982). On estimating the end-point of a distribution. Ann. Statist. 10 556–568.
  • Hall, P., Nussbaum, M. and Stern, S.E. (1997). On the estimation of a support curve of indeterminate sharpness. J. Multivariate Anal. 62 204–232.
  • Hall, P., Park, B.U. and Stern, S.E. (1998). On polynomial estimators of frontiers and boundaries. J. Multivariate Anal. 66 71–98.
  • Hall, P. and Park, B.U. (2004). Bandwidth choice for local polynomial estimation of smooth boundaries. J. Multivariate Anal. 91 240–261.
  • Hall, P. and Simar, L. (2002). Estimating a changepoint, boundary, or frontier in the presence of observation error. J. Amer. Statist. Assoc. 97 523–534.
  • Härdle, W., Park, B.U. and Tsybakov, A.B. (1995). Estimation of non-sharp support boundaries. J. Multivariate Anal. 55 205–218.
  • Harter, H.L. and Moore, A.H. (1965). Maximum-likelihood estimation of the parameters of gamma and Weilbull populations from complete and from censored samples. Technometrics 7 639–643.
  • Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 1163–1174.
  • Hirano, K. and Porter, J.R. (2003). Asymptotic efficiency in parametric structural models with parameter-dependent support. Econometrica 71 1307–1338.
  • Jofre-Bonet, M. and Pesendorfer, M. (2003). Estimation of a dynamic auction game. Econometrica 71 1443–1489.
  • Jurečková, J. (2000). Test of tails based on extreme regression quantiles. Statist. Probab. Lett. 49 53–61.
  • Kneip, A., Park, B.U. and Simar, L. (1998). A note on the convergence of nonparametric DEA estimators for production efficiency scores. Econometric Theory 14 783–793.
  • Knight, K. (2001a). Epi-convergence in distribution and stochastic equi-semicontinuity. Manuscript.
  • Knight, K. (2001b). Limiting distributions of linear programming estimators. Extremes 4 87–103.
  • Koenker, R., Ng, P. and Portnoy, S. (1994). Quantile smoothing splines. Biometrika 81 673–680.
  • Korostelev, A.P., Simar, L. and Tsybakov, A.B. (1995a). Efficient estimation of monotone boundaries. Ann. Statist. 23 476–489.
  • Korostelev, A.P., Simar, L. and Tsybakov, A.B. (1995b). On estimation of monotone and convex boundaries. Publ. Inst. Statist. Univ. Paris 39 3–18.
  • Korostelev, A.P. and Tsybakov, A.B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statistics 82. New York: Springer.
  • Long, S.J. (1997). Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, CA: Sage Publications.
  • Paarsch, H.J. (1992). Deciding between the common and private value paradigms in empirical models of auctions. J. Econometrics 51 191–215.
  • Park, B.U. and Simar, L. (1994). Efficient semiparametric estimation in a stochastic frontier model. J. Amer. Statist. Assoc. 89 929–936.
  • Portnoy, S. and Jurečková, J. (2000). On extreme regression quantiles. Extremes 2 227–243.
  • Robinson, W.T. and Chiang, J.W. (1996). Are Sutton’s predictions robust? Empirical insights into advertising, R&D, and concentration. J. Indust. Economics 44 389–408.
  • Smith, R.L. (1985). Maximum likelihood estimation in a class of nonregular cases. Biometrika 72 67–90.
  • Smith, R.L. (1994). Nonregular regression. Biometrika 81 173–183.
  • Zipf, G.K. (1941). National Unity and Disunity: The Nation as a Bio-Social Organism. Bloomington, IN: Principia Press.
  • Zipf, G.K. (1949). Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology. Cambridge, MA: Addison-Wesley.