Statistics and Computing

, Volume 17, Issue 4, pp 349–355 | Cite as

Frequent problems in calculating integrals and optimizing objective functions: a case study in density deconvolution



Many statistical procedures involve calculation of integrals or optimization (minimization or maximization) of some objective function. In practical implementation of these, the user often has to face specific problems such as seemingly numerical instability of the integral calculation, choices of grid points, appearance of several local minima or maxima, etc. In this paper we provide insights into these problems (why and when are they happening?), and give some guidelines of how to deal with them. Such problems are not new, neither are the ways to deal with them, but it is worthwhile to devote serious considerations to them. For a transparant and clear discussion of these issues, we focus on a particular statistical problem: nonparametric estimation of a density from a sample that contains measurement errors. The discussions and guidelines remain valid though in other contexts. In the density deconvolution setting, a kernel density estimator has been studied in detail in the literature. The estimator is consistent and fully data-driven procedures have been proposed. When implemented in practice however, the estimator can turn out to be very inaccurate if no adequate numerical procedures are used. We review the steps leading to the calculation of the estimator and in selecting parameters of the method, and discuss the various problems encountered in doing so.


Bandwidth selection Dramatic cancellation Fast Fourier transform Numerical approximations Optimization 


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  1. Carroll, R.J., Hall, P.: Optimal rates of convergence for deconvolving a density. J. Am. Stat. Assoc. 83, 1184–1186 (1988) MATHCrossRefGoogle Scholar
  2. Delaigle, A., Gijbels, I.: Estimation of integrated squared density derivatives from a contaminated sample. J. Roy. Stat. Soc. B 64, 869–886 (2002) MATHCrossRefGoogle Scholar
  3. Delaigle, A., Gijbels, I.: Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Ann. Inst. Stat. Math. 56, 19–47 (2004a) MATHCrossRefGoogle Scholar
  4. Delaigle, A., Gijbels, I.: Practical bandwidth selection in deconvolution kernel density estimation. Comput. Stat. Data Anal. 45, 249–267 (2004b) CrossRefGoogle Scholar
  5. Delaigle, A., Hall, P.: On the optimal kernel choice for deconvolution. Stat. Probab. Lett. 76, 1594–1602 (2006) MATHCrossRefGoogle Scholar
  6. Fan, J.: Asymptotic normality for deconvolution kernel density estimators. Sankhya A 53, 97–110 (1991a) MATHGoogle Scholar
  7. Fan, J.: Global behaviour of deconvolution kernel estimates. Stat. Sinica 1, 541–551 (1991b) MATHGoogle Scholar
  8. Fan, J.: On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Stat. 19, 1257–1272 (1991c) MATHGoogle Scholar
  9. Fan, J.: Deconvolution with supersmooth distributions. Can. J. Stat. 20, 155–169 (1992) MATHCrossRefGoogle Scholar
  10. Hesse, C.H.: Data-driven deconvolution. Nonparametr. Stat. 10, 343–373 (1999) MATHCrossRefGoogle Scholar
  11. Masry, E.: Asymptotic normality for deconvolution estimators of multivariate densities of stationary processes. J. Multivar. Anal. 44, 47–68 (1993a) MATHCrossRefGoogle Scholar
  12. Masry, E.: Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stoch. Process. Appl. 47, 53–74 (1993b) MATHCrossRefGoogle Scholar
  13. Meister, A.: On the effect of misspecifying the error density in a deconvolution problem. Can. J. Stat. 32, 439–449 (2004) MATHGoogle Scholar
  14. Meister, A.: Density estimation with normal measurement error with unknown variance. Stat. Sinica 16, 195–211 (2006) MATHGoogle Scholar
  15. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in C, The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992) MATHGoogle Scholar
  16. Stefanski, L., Carroll, R.J.: Deconvoluting kernel density estimators. Statistics 2, 169–184 (1990) CrossRefGoogle Scholar
  17. Van Es, A.J., Uh, H.-W.: Asymptotic normality of kernel type deconvolution estimators: crossing the Cauchy boundary. Nonparametr. Stat. 16, 261–277 (2004) MATHCrossRefGoogle Scholar
  18. Van Es, A.J., Uh, H.-W.: Asymptotic normality of kernel type deconvolution estimators. Scand. J. Stat. 32, 467–483 (2005) CrossRefGoogle Scholar
  19. Zhang, S., Karunamuni, R.: Boundary bias correction for nonparametric deconvolution. Ann. Inst. Stat. Math. 52, 612–629 (2000) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BristolBristolUK
  2. 2.Department of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  3. 3.Department of Mathematics and University Center for StatisticsKatholieke Universiteit LeuvenHeverlee (Leuven)Belgium

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