International Journal of Nonlinear Analysis and Applications

Block thresholding in wavelet estimation of the regression function with errors in variables

Document Type : Research Paper

Author

Department of Statistics, Faculty of Science, Gonbad Kavous University, Gonbad Kavous 4971799151, Iran

Abstract

Errors-in-variables regression is the study of the association between covariates and responses where covariates are observed with errors. In this paper, we consider the estimation of regression functions when the independent variable is measured with error. We investigate the performances of an adaptive wavelet block thresholding estimator via the minimax approach under the $L_p$ risk with $p \geq 1$ over Besov balls. We prove that it achieves the optimal rates of convergence.

Keywords

[1] F. Comte and M.-L. Taupin, Nonparametric estimation of the regression function in an errors-in-variables model, Statistica Sinica 17 (2007), no. 3, 1065-1090.
[2] C.-L. Cheng and H. Schneeweiss, Polynomial regression with errors in the variables, J. Roy. Stat. Soc. Ser. B Stat. Methodol. 60 (1998), 189—199.
[3] C. Chesneau, On adaptive wavelet estimation of the regression function and its derivatives in an errors-in-variables model, Current Dev. Theory Appl. Wavelets 4 (2010), 185–208.
[4] D.L. Donoho and I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, J. Amer. Statist. Assoc. 90 (1995), no. 432, 1200–1224.
[5] J. Fan and J.Y. Koo, Wavelet deconvolution, IEEE Trans. Inf. Theory 48 (2002), 734—747.
[6] J. Fan and M. Masry, Multivariate regression estimation with errors-in-variables: Asymptotic normality for mixing process, J. Multivariate Anal. 43 (1992), 237–272.
[7] J. Fan and Y.K. Truong, Nonparametric regression with errors-in-variables, Ann. Statist. 21 (1993), no. 4, 1900–1925.
[8] J. Fan, Y.K. Truong and Y. Wang, Nonparametric function estimation involving errors-in-variables, Springer, 1991.
[9] W. Hardle, G. Kerkyacharian, D. Picard and A. Tsybakov, Wavelet, Approximation and Statistical Applications, Lectures Notes in Statistics, Springer Verlag, New York, 1998.
[10] P. Hall, G. Kerkyacharian D. Picard, On the minimax optimality of block thresholded wavelet estimators, Ann. Statist. 9 (1999), 33—50.
[11] D.A. Ioannides and P.D. Alevizos, Nonparametric regression with errors in variables and applications, Statist. Probab. Lett. 32 (1997), 35–43.
[12] I.M. Johnstone, G. Kerkyacharian, D. Picard and M. Raimondo, Wavelet deconvolution in a periodic setting, J. Roy. Statist. Soc. Ser. B 66 (2004), no. 3, 547—573.
[13] M.C. Jones, Estimating densities, quantiles, quantile densities and density quantiles, Ann. Inst. Statist. Math. 44 (1992), no. 4, 721–727.
[14] K. Kato and Y. Sasaki, Uniform confidence bands for nonparametric errors-in-variables regression, J. Econ. 213 (2019), no. 2, 516–555.
[15] J.Y. Koo and K.W. Lee, B-splines estimations of regression functions with errors in variables, Statist. Probab. Lett. 40 (1998), 57–66.
[16] M. Pensky and B. Vidakovic, Adaptive wavelet estimator for nonparametric density deconvolution, Ann. Statist. 27 (1999), 2033—2053.
[17] E. Masry, Multivariate regression estimation with errors-in-variables for stationary processes, Nonparamet. Statist. 3 (1993), 13–36.
[18] S. Mallat, A wavelet tour of signal processing, Elsevier/ Academic Press, Amsterdam, 2009.
[19] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992
[20] N.U. Nair, P.G. Sankaran and B.V. Kumar, Total time on test transforms of order n and its implications in reliability analysis, J. Appl. Probab. 45 (2008), 1126–1139.
[21] N.U. Nair and P.G. Sankaran, Quantile based reliability analysis, Commun. Statist.: Theory Meth. 38 (2009), 222–232.
[22] E. Parzen, Non parametric statistical data modeling, J. Amer. Statist. Assoc. 74 (1979), 105–122.
[23] L. Peng and J.P. Fine, Nonparametric quantile inference with competing risks data, Biometrika 94 (2007), 735–744.
[24] N. Reid, Estimating the median survival time, Biometrika 68 (1981), 601–608.
[25] P.G. Sankaran and N.U. Nair, Nonparametric estimation of hazard quantile function, J. Nonparamet. Statist. 21 (2009), 757–767.
[26] P.G. Sankaran, I. Dewan and E.P. Sreedevi, A non-parametric test for stochastic dominance using total time on test transform, Amer. J. Math. Manag. Sci. 34 (2015), 162–183
[27] E. Shirazi, Y. Chaubey, H. Doosti, H.A. Nirumand, Wavelet-based estimation for the derivative of a density by block thresholding under random censorship, J. Korean Statist. Soc. 41 (2012), 199—211.
[28] E. Shirazi, H. Doosti, H.A. Niroumand and N. Hosseinioun, Nonparametric regression estimate with censored data based on block thresholding method, J. Statist. Plan. Infer., 143 (2013), 1150–1165.
[29] E.V. Slud, D.P. Byar and S.B. Green, A comparison of reflected versus test-based 355 confidence intervals for the median survival time, based on censored data, Biometrics 40 (1984), 587–600.
[30] P. Soni, I. Dewan and K. Jain, Nonparametric estimation of quantile density function, Comput. Statist. Data Anal. 56 (2012), no. 12, 3876–3886.
[31] J.Q. Su and L.J. Wei, Nonparametric estimation for the difference or ratio of median failure times, Biometrics 49 (1993), 603—607.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 9
September 2023
Pages 367-377
  • Receive Date: 16 March 2022
  • Revise Date: 05 July 2022
  • Accept Date: 10 September 2022