## Brazilian Journal of Probability and Statistics

### Wavelet estimation for derivative of a density in the presence of additive noise

B. L. S. Prakasa Rao

#### Abstract

We construct a wavelet estimator for the derivative of a probability density function in the presence of an additive noise and study its $L_{p}$-consistency property.

#### Article information

Source
Braz. J. Probab. Stat., Volume 32, Number 4 (2018), 834-850.

Dates
Accepted: July 2017
First available in Project Euclid: 17 August 2018

https://projecteuclid.org/euclid.bjps/1534492904

Digital Object Identifier
doi:10.1214/17-BJPS369

Mathematical Reviews number (MathSciNet)
MR3845032

Zentralblatt MATH identifier
06979603

#### Citation

Prakasa Rao, B. L. S. Wavelet estimation for derivative of a density in the presence of additive noise. Braz. J. Probab. Stat. 32 (2018), no. 4, 834--850. doi:10.1214/17-BJPS369. https://projecteuclid.org/euclid.bjps/1534492904

#### References

• Andersen, K. and Hansen, M. (2001). Multiplicative censoring: Density estimation by a series expansion. Journal of Statistical Planning and Inference 98, 137–155.
• Antoniadis, A., Gregoire, G. and McKeague, I. W. (1994). Wavelet methods for curve estimation. Journal of the American Statistical Association 89, 1340–1353.
• Asgharian, M., Carone, M. and Fakoor, V. (2012). Large-sample study of the kernel density estimation under multiplicative censoring. The Annals of Statistics 40, 159–187.
• Chaubey, Y. P., Chesneau, C. and Doosti, H. (2011). On linear wavelet density estimation: Some recent developments. Journal of the Indian Society of Agricultural Statistics 65, 169–179.
• Chaubey, Y. P., Chesneau, C. and Doosti, H. (2015). Adaptive wavelet estimation of a density from mixtures under multiplicative censoring. Statistics 49, 638–659.
• Chaubey, Y. P., Doosti, H. and Prakasa Rao, B. L. S. (2006). Wavelet based estimation of the derivatives of a density with associated variables. International Journal of Pure and Applied Mathematics 27, 97–106.
• Chaubey, Y. P., Doosti, H. and Prakasa Rao, B. L. S. (2008). Wavelet based estimation of the derivatives of a density for a negatively associated process. Journal of Statistical Theory and Practice 2, 453–463.
• Chesneau, C. (2013). Wavelet estimation of a density in a GARCH-type model. Communications in Statistics Theory and Methods 42, 98–117.
• Chesneau, C. and Doosti, H. (2012). Wavelet linear density estimation for a GARCH model under various dependence structures. Journal of the Iranian Statistical Society 11, 1–21.
• Daubechies, I. (1988). Orthogonal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics 41, 909–996.
• Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia: SIAM.
• Donoho, D., Johnstone, I., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. The Annals of Statistics 24, 508–539.
• Fan, J. and Koo, J. (2002). Wavelet deconvolution. IEEE Transactions on Information Theory 48, 734–747.
• Geng, Z. and Wang, J. (2015). The mean consistency of wavelet density estimators. Journal of Inequalities and Applications 2015, Article ID 111.
• Hardle, W., Kerkycharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximations, and Statistical Applications. Lecture Notes in Statistics 129. New York: Springer.
• Hosseinioun, N. (2016). Wavelet-based density estimation in presence of additive noise under various dependence structures. Advances in Pure Mathematics 6, 7–15.
• Leblanc, F. (1996). Wavelet linear density estimator for a discrete-time stochastic process: $L_{p}$-losses. Statistics & Probability Letters 27, 71–84.
• Li, R. and Liu, Y. (2014). Wavelet estimations for a density with some additive noises. Applied and Computational Harmonic Analysis 36, 416–433.
• Li, Q. and Racine, J. S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton: Princeton University Press.
• Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. The Annals of Statistics 39, 201–231.
• Mallat, S. G. (1989). A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 674–693.
• Prakasa Rao, B. L. S. (1983). Nonparametric Functional Estimation. Orlando: Academic Press.
• Prakasa Rao, B. L. S. (1996). Nonparametric estimation of the derivatives of a density by the method of wavelets. Bulletin of Informatics and Cybernetics 28, 91–100.
• Prakasa Rao, B. L. S. (1999a). Nonparametric functional estimation: An overview. In Asymptotics: Nonparametrics and Time Series (S. Ghosh, ed.) 461–509. New York: Marcel Dekker Inc.
• Prakasa Rao, B. L. S. (1999b). Estimation of the integrated squared density derivative by wavelets. Bulletin of Informatics and Cybernetics 31, 47–65.
• Prakasa Rao, B. L. S. (2000). Nonparametric estimation of partial derivatives of a multivariate probability density by the method of wavelets. In Asymptotics in Statistics and Probability: Festschrift for George G. Roussas (M. L. Puri, ed.) 321–330. Amsterdam: VSP.
• Prakasa Rao, B. L. S. (2003). Wavelet linear density estimation for associated sequences. Journal of the Indian Statistical Association 41, 369–379.
• Prakasa Rao, B. L. S. (2010). Wavelet linear estimation for derivatives of a density from observations of mixtures with varying mixing proportions. Indian Journal of Pure and Applied Mathematics 41, 275–291.
• Prakasa Rao, B. L. S. (2017). Wavelet estimation for derivative of a density in a GARCH-type model. Communications in Statistics Theory and Methods 46, 2396–2410.
• Rosenthal, H. P. (1970). On the subspaces of $L^{p}$, ($p>2$) spanned by sequences of independent random variables. Israel Journal of Mathematics 8, 273–303.
• Shirazi, E., Chaubey, Y. P., Doosti, H. and Nirumand, H. A. (2012). Wavelet based estimation for the derivative of a density by block thresholding under random censorship. Journal of the Korean Statistical Society 41, 199–211.
• Strang, G. (1989). Wavelets and dilation equations: A brief introduction. SIAM Review 31, 614–627.
• Tribouley, K. (1995). Practical estimation of multivariate densities using wavelet methods. Statistica Neerlandica 49, 41–62.
• Vardi, Y. (1989). Multiplicative censoring, renewal processes, deconvolution and decreasing density: Nonparmetric estimation. Biometrika 76, 751–761.
• Vardi, Y. and Zhang, C. H. (1992). Large sample study of empirical distributions in a random multiplicative censoring model. The Annals of Statistics 20, 1022–1039.
• Vidakovic, B. (1999). Statistical Modeling by Wavelets. New York: Wiley.
• Walter, G. and Ghorai, J. (1992). Advantages and disadvantages of density estimation with wavelets. In Graphics and Visualization: Proceedings of the 24th Symposium on the Interface (H. J. Newton, ed.). Computing Science and Statistics 24, 234–243. Fairfax Station: Interface Foundation of North America.