Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

A Note on Deconvolution Estimators when Measurement Errors are Normal

  • Received : 2012.03.22
  • Accepted : 2012.06.20
  • Published : 2012.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper a support vector method is proposed for use when the sample observations are contaminated by a normally distributed measurement error. The performance of deconvolution density estimators based on the support vector method is explored and compared with kernel density estimators by means of a simulation study. An interesting result was that for the estimation of kurtotic density, the support vector deconvolution estimator with a Gaussian kernel showed a better performance than the classical deconvolution kernel estimator.

Keywords

References

  1. Aronszajn, N. (1950). Theory of reproducing kernels, Transactions of the American Mathematical Society, 68, 337-404. https://doi.org/10.1090/S0002-9947-1950-0051437-7
  2. Bochner, S. (1959). Lectures on Fourier Integral, Princeton University Press, Princeton, New Jersey.
  3. Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvoluting a density, Journal of the American Statistical Association, 83, 1184-1886. https://doi.org/10.1080/01621459.1988.10478718
  4. Delaigle, A. and Gijbels, I. (2007). Frequent problems in calculating integrals and optimizing objective functions: A case study in density estimation, Statistics and Computing, 17, 349-355. https://doi.org/10.1007/s11222-007-9024-0
  5. Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problem, Annals of Statistics, 19, 1257-1272. https://doi.org/10.1214/aos/1176348248
  6. Fan, J. (1992). Deconvolution with supersmooth distribution, The Canadian Journal of Statistics, 20, 159-169.
  7. Gunn, S. R. (1998). Support Vector Machines for Classification and Regression, Technical report, University of Southampton.
  8. Hall, P. and Qiu, P. (2005). Discrete-transform approach to deconvolution problems, Biometrika, 92, 135-148. https://doi.org/10.1093/biomet/92.1.135
  9. Hazelton, M. L. and Turlach, B. A. (2009). Nonparametric density deconvolution by weighted kernel estimators, Statistics and Computing, 19, 217-228. https://doi.org/10.1007/s11222-008-9086-7
  10. Lee, S. (2010). A support vector method for the deconvolution problem, Communications of the Korean Statistical Society, 17, 451-457. https://doi.org/10.5351/CKSS.2010.17.3.451
  11. Lee, S. and Taylor, R. L. (2008). A note on support vector density estimation for the deconvolution problem, Communications in Statistics: Theory and Methods, 37, 328-336. https://doi.org/10.1080/03610920701653086
  12. Liu, M. C. and Taylor, R. L. (1989). A Consistent nonparametric density estimator for the deconvolution problem, The Canadian Journal of Statistics, 17, 427-438. https://doi.org/10.2307/3315482
  13. Mendelsohn, J. and Rice, R. (1982), Deconvolution of microfluorometric histograms with B splines, Journal of the American Statistical Association, 77, 748-753.
  14. Mukherjee, S. and Vapnik, V. (1999). Support Vector Method for Multivariate Density Estimation, Technical Report A.I. Memo no. 1653, MIT AI Lab.
  15. Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolutoin, Annals of Statistics, 27, 2033-2053. https://doi.org/10.1214/aos/1017939249
  16. Phillips, D. L. (1962). A technique for the numerical solution of integral equations of the first kind, Journal of the Association for Computing Machinery, 9, 84-97. https://doi.org/10.1145/321105.321114
  17. Stefanski, L. and Carroll, R. J. (1990). Deconvoluting kernel density estimators, Statistics, 21, 169-184. https://doi.org/10.1080/02331889008802238
  18. Vapnik, V. (1995). The Nature of Statistical Learning Theory, Springer Verlag, New York.
  19. Vapnik, V. and Chervonenkis, A. (1964). A note on one class of perceptrons, Automation and Remote Control, 25, 103-109.
  20. Vapnik, V. and Lerner, L. (1963). Pattern Recognition using generalized portrait method, Automation and Remote Control, 24.
  21. Vert, R. and Vert, J. (2006). Consistency and convergence rates of one-class svms and related algorithms, Journal of Machine Learning Research, 7, 817-854.
  22. Wand, M. P. (1998). Finite sample performance of deconvolving density estimators, Statistics and Probability Letters, 37, 131-139. https://doi.org/10.1016/S0167-7152(97)00110-7
  23. Weston, J., Gammerman, A., Stitson, M., Vapnik, V., Vovk, V. and Watkins, C. (1999). Support vector density estimation. In Scholkopf, B. and Smola, A., editors, Advances in Kernel Methods-Suppot Vector Learning, 293-306, MIT Press, Cambridge, MA.
  24. Zhang, H. P. (1992). On deconvolution using time of flight information in positron emission tomography, Statistica Sinica, 2, 553-575.

Cited by

  1. A note on nonparametric density deconvolution by weighted kernel estimators vol.25, pp.4, 2014, https://doi.org/10.7465/jkdi.2014.25.4.951
  2. A note on SVM estimators in RKHS for the deconvolution problem vol.23, pp.1, 2016, https://doi.org/10.5351/CSAM.2016.23.1.071