Wavelet- Threshold Nonparametric Density Estimator and Covariance Structure of Wavelet Coefficients

Document Type : Scientific Research

Authors

1 Associate Professor, Department of Statistics, Persian Gulf University, Bushehr

2 Associate Professor, Department of Statistics, Payam Noor University

3 M.Sc., Department of Statistics, Payam Noor University

Abstract

Wavelets are one of the newest achievements of mathematical science, which have many applications in other sciences especially statistics. In this paper, after introducing wavelet transforms, the nonparametric estimator of the density function is expressed by the nuclear wavelet and threshold wavelet method. Also variance-covariance of wavelet coefficient are investigated. At the end we survey the theoretical outcomes with numerical computation by using R software to compare purpose estimators.

Keywords

References

[1] Afshari, M. (2013). A fast wavelet algorithm for analyzing one-dimensional signal processing and asymptotic distribution of wavelet coefficients with numerical example and simulation. Communications in Statistics-Theory and Methods, 42(22), 4156-4169.
[2] Afshari, M. (2014). Wavelet density estimation of censoring data and evaluate of mean integral square error with convergence ratio and empirical distribution of given estimator. Applied Mathematics, 5(13), 20-62.
[3] Afshari, M., Lak, F., & Gholizadeh, B. (2016). A new Bayesian wavelet thresholding estimator of nonparametric regression. Journal of Applied Statistics, 1-18.
[4] Afshari, M. (2017). Nonlinear wavelet Shrinkage Estimator of Nonparametric Regression Function Via Cross Validation. International Journal of Wavelets, Multiresolution and Information, 15(6}, 1-16.
[5] Antoniadis, A., Gregoire, G. and Mckeague, I. (1994). Wavelet methods for curve estimation. Journal of the Americal Statistical Association 89: 1340-1353.
[6] Donoho, D.L., Johnstone, I.M., Krekyacharian, G., & Picard, D. (1996). Density Estimation by Wavelet Thresholding. The Annals of Statistics, 24(2), 508-539.
[7] Daubechies, I. (1986). Orthonormal bases of compactly supported wavelets. Communications on pure and applied mathematics, 41(7), 909-996.
[8] Doosti, H., H. A. Niroumand, and M. Afshari. (2006). Wavelet Based Estimation of the Derivatives of a Density for a Discrete-Time Stochastic Process: Lp-losses. Journal of Sciences, Islamic Republic of Iran 17(1), 75-81.
[9] Doosti, H., Afshari, M., & Niroumand, H. A. (2008). Wavelets for nonparametric stochastic regression with mixing stochastic process. Communications in Statistics — Theory and Methods, 37(3), 373-385.
[10]Grossmann, A., Morlet, J. (1984). Decomposition of Hardly functions into square integrable wavelets of constant shape. SIAM J. Math.15:723-736.
[11] Haar, A. (1910). Zur thorie der orthogonal Functioned - system, Mathematics annalysis, 69: 331-371.
[12] Herrick, D.R.M. (2000). Wavelet Methods for Curve and Surface Estimation. Ph.D. thesis, University of Bristol.
Grossmann, A., Morlet, J. (1984). Decomposition of Hardly functions into square integrable wavelets of constant shape. SIAM J. Math.15:723-736.
[13] Kovac, A. and Silverman, B. W. (2000). Extending the scope of wavelet regression methods by coefficient-dependent thresholding. J. Am. Statist.
[14] Mallat, S. (1989). Multiresolution approximations and wavelet orthonormal bases of L2(R). Trans. Amer. Math. Soc. 315, 69-87.
[15] Nason, G. (2011). Wavelet Methods in Statistic with R.
[16] Vidakovic, B. (1998). Nonlinear wavelet shrinkage with Bayes rules and Bayes factors. Journal of the American Statistical Association, 93(441), 173-179.

Files

Share

How to cite

Statistics