Journal of Statistical Theory and Practice

, Volume 5, Issue 4, pp 697–714 | Cite as

Data Driven Smooth Test for Contaminated Data

  • D. PommeretEmail author


In this paper we consider a random variable Y contaminated by an additive noise Z from a known distribution. Our purpose is to test the distribution of the unobserved random variable Y. We propose a data driven statistic based on a nonparametric expansion of the density of Y +Z, which can be applied as well in the continuous case as in the discrete case. The problem is considered at first in the univariate case, and then extended in a multivariate setting with a bootstrap procedure. Finite-sample properties are examined through Monte Carlo and Quasi Monte Carlo simulations in univariate and bivariate cases.

AMS Subject Classification

62G10 62F05 


Bootstrap Contaminated data Data driven test Multivariate sample Orthogonal polynomials Quasi Monte Carlo 


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  1. Abramowitz, M., Stegun, I.A., 1972. Orthogonal polynomials. In Handbook of Mathematical Functions, Abramowitz, M. and Stegun, I.A. (Editors), 9th printing, Chap. 22, Dover, New York.zbMATHGoogle Scholar
  2. Ash, R.B., Gardner, M.F., 1975. Topics in stochastic processes. Probability and Mathematical Statistics, Academic Press, New York.Google Scholar
  3. Bissantz, N., Du¨mbgen, L., Holzmann, H., Munk, A., 2007. Non-parametric confidence bands in deconvolution density estimation. J. Roy. Stat. Soc. B, 69, 483–506.MathSciNetCrossRefGoogle Scholar
  4. Carroll, R.J., Hall, P., 1988. Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc., 83, 1184–1186.MathSciNetCrossRefGoogle Scholar
  5. Delaigle, A., Gijbels, I., 2004. Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Ann. Inst. Statist. Math., 56, 19–47.MathSciNetCrossRefGoogle Scholar
  6. Devroye, L., 1989. Consistent deconvolution in density estimation. Canad. J. Statist., 17, 235–239.MathSciNetCrossRefGoogle Scholar
  7. Eagleson, G., 1964. Polynomial expansion of bivariate distribution. Ann. Math. Statist., 35, 1208–1215.MathSciNetCrossRefGoogle Scholar
  8. L’Ecuyer, P., 2006. Uniform random number generation. In Handbook in OR & MS, Hendersonm, S.G. and Nelson B.L. (Editors), Vol. 13, Chap. 3, Elsevier.Google Scholar
  9. Fan, J., 1991. On the optimal rates of convergence for nonparametric deconvolution problems, Ann. Statist., 19, 1257–1272.MathSciNetCrossRefGoogle Scholar
  10. Ghattas, B., Pommeret, D., Reboul, L., Yao, A.F., 2011. Data driven smooth test for paired populations. Journal of Statistical Planning and Inference, 141, 262–275.MathSciNetCrossRefGoogle Scholar
  11. Holzmann, H., Bissantz, N., Munk, A., 2007. Density testing in a contaminated sample. J. Multiv. Anal., 98, 57–75.MathSciNetCrossRefGoogle Scholar
  12. Koudou, A.E., 1996. Probabilités de Lancaster. Exp. Math., 14, 247–275.MathSciNetzbMATHGoogle Scholar
  13. Langovoy, M.A., 2008. Data-driven efficient score tests for deconvolution hypotheses. Inverse Problem, 24, 25–28.MathSciNetCrossRefGoogle Scholar
  14. Ledwina, T., 1994. Data-driven version of neymans smooth test of fit. J. Amer. Statist. Assoc., 89, 1000–1005.MathSciNetCrossRefGoogle Scholar
  15. Rayner, J.C.W., Best, D.J., 1989. Smooth Tests of Goodness of Fit. Oxford University Press, New York.zbMATHGoogle Scholar
  16. Zhang, C.H., 1990. Fourier methods for estimating mixing densities and distributions. Ann. Statist., 18, 806–831.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2011

Authors and Affiliations

  1. 1.Institut de Mathématiques de Luminy - Case 907Université de la MéditerranéeFrance

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