The Annals of Statistics

On the minimisation of $L\sp p$ error in mode estimation

Birgit Grund and Peter Hall

Full-text: Open access


We show that, for $L^p$ convergence of the mode of a nonparametric density estimator to the mode of an unknown probability density, finiteness of the pth moment of the underlying distribution is both necessary and sufficient. The basic requirement of existence of finite variance has been overlooked by statisticians, who have earlier considered mean square convergence of nonparametric mode estimators; they have focussed on mean squared error of the asymptotic distribution, rather than on asymptotic mean squared error. The effect of bandwidth choice on the rate of $L^p$ convergence is analysed, and smoothed bootstrap methods are used to develop an empirical approximation to the $L^p$ measure of error. The resulting bootstrap estimator of $L^p$ error may be minimised with respect to the bandwidth of the nonparametric density estimator, and in this way an empirical rule may be developed for selecting the bandwidth for mode estimation. Particular attention is devoted to the problem of selecting the appropriate amount of smoothing in the bootstrap algorithm.

Article information

Ann. Statist., Volume 23, Number 6 (1995), 2264-2284.

First available in Project Euclid: 15 October 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Bandwidth bootstrap convergence in $L^p$ kernel density estimator mean squared error mode smoothed bootstrap smoothing parameter


Grund, Birgit; Hall, Peter. On the minimisation of $L\sp p$ error in mode estimation. Ann. Statist. 23 (1995), no. 6, 2264--2284. doi:10.1214/aos/1034713656.

Export citation


  • BHATTACHARy A, R. N. and RAO, R. R. 1976. Normal Approximation and Asy mptotic Expansions. Wiley, New York. Z. 1
  • DEVROy E, L. 1985. Nonparametric Density Estimation: The L View. Wiley, New York. Z.
  • EDDY, W. 1980. Optimal kernel estimators of the mode. Ann. Statist. 8 870 882. Z.
  • EDDY, W. 1982. The asy mptotic distributions of kernel estimators of the mode. Z. Wahrsch. Verw. Gebiete 59 279 290. Z.
  • FARAWAY, J. J. and JHUN, M. 1990. Bootstrap choice of bandwidth for density estimation. J. Amer. Statist. Assoc. 85 1119 1122. Z.
  • FISHER, N. I., MAMMEN, E. and MARRON, J. S. 1994. Testing for multimodality. Comput. Statist. Data Anal. 18 499 512. Z.
  • GARSIA, A. M. 1970. Continuity properties of Gaussian processes with multidimensional time parameter. Proc. Sixth Berkeley Sy mp. Math. Statist. Probab. 2 269 274. Univ. California Press, Berkeley. Z.
  • GRENANDER, U. 1965. Some direct estimates of the mode. Ann. Math. Statist. 36 131 138. Z.
  • HALL, P. 1990. Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems. J. Multivariate Anal. 32 177 203. Z.
  • HALL, P., MARRON, J. S. and PARK, B. U. 1992. Smoothed cross-validation. Probab. Theory Related Fields 92 1 20. Z.
  • HALL, P. and MURISON, R. D. 1991. Correcting the negativity of high-order kernel density estimators, Report CMA-SR21-91, Centre for Mathematics and its Applications, Australian National Univ. Z.
  • KOMLOS, J., MAJOR, P. and TUSNADY, G. 1975. An approximation of partial sums of independent ´ ´ rv's and the sample df. I. Z. Wahrsch. Verw. Gebiete 32 111 131. Z.
  • MAMMEN, E., MARRON, J. S. and FISHER, N. I. 1992. Some asy mptotics for multimodality tests based on kernel estimates. Probab. Theory Related Fields 91 115 132. Z.
  • MULLER, H.-G. 1989. Adaptive nonparametric peak regression. Ann. Statist. 17 1053 1069. ¨ Z.
  • PARZEN, E. 1962. On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065 1076.
  • ROMANO, J. P. 1988a. On weak convergence and optimality of kernel density estimates of the mode. Ann. Statist. 16 629 647. Z.
  • ROMANO, J. P. 1988b. Bootstrapping the mode. Ann. Inst. Statist. Math. 40 565 586. Z.
  • SILVERMAN, B. W. 1978. Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist. 6 177 184. Z.
  • SILVERMAN, B. W. 1986. Density Estimation for Statistics and Data Analy sis. Chapman and Hall, London. Z.
  • TAy LOR, C. C. 1989. Boostrap choice of the smoothing parameter in kernel density estimation. Biometrika 76 705 712. Z.
  • TSy BAKOV, A. B. 1990. Recursive estimation of the mode of a multivariate distribution. Problems Inform. Transmission 26 31 37.