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Wavelet-Based Density Estimation in Presence of Additive Noise under Various Dependence Structures

DOI: 10.4236/apm.2016.61002, PP. 7-15

Keywords: Additive Noise, Density Estimation, Dependent Sequence, Rate of Convergence, Wavelet

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Abstract:

We study the following model: \"\". The aim is to estimate the distribution of X when only \"\"?are observed. In the classical model, the distribution of \"\"?is assumed to be known, and this is often considered as an important drawback of this simple model. Indeed, in most practical applications, the distribution of the errors cannot be perfectly known. In this paper, the author will construct wavelet estimators and analyze their asymptotic mean integrated squared error for additive noise models under certain dependent conditions, the strong mixing case, the β-mixing case and the ρ-mixing case. Under mild conditions on the family of wavelets, the estimator is shown to be \"\"-consistent and fast rates of convergence have been established.

References

[1]  Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006) Penalized Contrast Estimator for Density Deconvolution. The Canadian Journal of Statistics, 34, 431-452.
http://dx.doi.org/10.1002/cjs.5550340305
[2]  Fan, J. and Koo, J.Y. (2002) Wavelet Deconvolution. IEEE Transactions on Information Theory, 48, 734-747.
http://dx.doi.org/10.1109/18.986021
[3]  Caroll, R.J. and Hall, P. (1988) Optimal Rates of Convergence for Deconvolving a Density. Journal of the American Statistical Association, 83, 1184-1186.
http://dx.doi.org/10.1080/01621459.1988.10478718
[4]  Johannes, J. (2009) Deconvolution with Unknown Error Distribution. Annals of Statistics, 37, 2301-2323.
http://dx.doi.org/10.1214/08-AOS652
[5]  Meister, A. (2004) On the Effect of Misspecifying the Error Density in a Deconvolution. Canadian Journal of Statistics, 32, 439-449.
[6]  Li, T. and Vuong, Q. (1998) Nonparametric Estimation of the Measurement Error Model Using Multiple Indicators. Journal of Multivariate Analysis, 65, 139-165.
http://dx.doi.org/10.1006/jmva.1998.1741
[7]  Delaigle, A., Hall, P. and Meister, A. (2008) On Deconvolution with Repeated Measurements. Annals of Statistics, 36, 665-685.
http://dx.doi.org/10.1214/009053607000000884
[8]  Neumann, M.H. (2007) Deconvolution from Panel Data with Unknown Error Distribution. Journal of Multivariate Analysis, 98, 1955-1968.
http://dx.doi.org/10.1016/j.jmva.2006.09.012
[9]  Odiachi, P. and Prieve, D. (2004) Removing the Effects of Additive Noise in Term. Journal of Colloid and Interface Science, 270, 113-122.
http://dx.doi.org/10.1016/S0021-9797(03)00548-4
[10]  Devroye, L. (1989) Consistent Deconvolution in Density Estimation. Canadian Journal of Statistics, 17, 235-239.
http://dx.doi.org/10.2307/3314852
[11]  Fan, J. (1991) On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems. Annals of Statistics, 19, 1257-1272.
http://dx.doi.org/10.1214/aos/1176348248
[12]  Liu, M.C. and Taylor, R.L. (1989) A Consistent Nonparametric Density Estimator for the Deconvolution Problem. Canadian Journal of Statistics, 17, 427-438.
http://dx.doi.org/10.2307/3315482
[13]  Masry, E. (1991) Multivariate Probability Density Deconvolution for Stationary Random Processes. IEEE Transactions on Information Theory, 37, 1105-1115.
http://dx.doi.org/10.1109/18.87002
[14]  Stefanski, L. and Carroll, R.J. (1990) Deconvoluting Kernel Density Estimators. Statistics, 21, 169-184.
http://dx.doi.org/10.1080/02331889008802238
[15]  Zhang, C.-H. (1990) Fourier Methods for Estimating Mixing Densities and Distributions. Annals of Statistics, 18, 806-831.
http://dx.doi.org/10.1214/aos/1176347627
[16]  Hesse, C.H. (1999) Data-Driven Deconvolution. Journal of Nonparametric Statistics, 10, 343-373.
http://dx.doi.org/10.1080/10485259908832766
[17]  Cator, E.A. (2001) Deconvolution with Arbitrarily Smooth Kernels. Statistics & Probability Letters, 54, Article ID: 205214.
http://dx.doi.org/10.1016/s0167-7152(01)00083-9
[18]  Delaigle, A. and Gijbels, I. (2004) Bootstrap Bandwidth Selection in Kernel Density Estimation from a Contaminated Sample. Annals of the Institute of Statistical Mathematics, 56, 1947.
http://dx.doi.org/10.1007/BF02530523
[19]  Koo, J.-Y. (1999) Logspline Deconvolution in Besov Space. Scandinavian Journal of Statistics, 26, 73-86.
http://dx.doi.org/10.1111/1467-9469.00138
[20]  Efromovich, S. (1997) Density Estimation for the Case of Super-Smooth Measurement Error. Journal of the American Statistical Association, 92, 526-535.
http://dx.doi.org/10.1080/01621459.1997.10474005
[21]  Geng, Z. and Wang, J. (2015) The Mean Consistency of Wavelet Density Estimators. Journal of Inequalities and Applications, 2015, 111.
http://dx.doi.org/10.1186/s13660-015-0636-1
[22]  Daubechies, I. (1992) Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics.
http://dx.doi.org/10.1137/1.9781611970104
[23]  Meyer, Y. (1992) Wavelets and Operators. Cambridge University Press, Cambridge.
[24]  Hardle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1997) Wavelets, Approximation and Statistical Applications. Springer, New York.
[25]  Tsybakov, A.B. (2009) Introduction to Nonparametric Estimation. Springer, Berlin.
[26]  Pensky, M. and Vidakovic, B. (1999) Adaptive Wavelet Estimator for Nonparametric Density Deconvolution. The Annals of Statistics, 27, 2033-2053.
[27]  Lounici, K. and Nickl, R. (2011) Global Uniform Risk Bounds for Wavelet Deconvolution Estimators. The Annals of Statistics, 39, 201-231.
http://dx.doi.org/10.1214/10-AOS836
[28]  Li, R. and Liu, Y. (2014) Wavelet Optimal Estimations for a Density with Some Additive Noises. Applied and Computational Harmonic Analysis, 36, 2.
[29]  Doukhan, P. (1994) Mixing. Properties and Examples. Lecture Notes in Statistics, 85, Springer Verlag, New York.
[30]  Antoniadis, A. (1997) Wavelets in Statistics: A Review (with Discussion). Journal of the Italian Statistical Society, Series B, 6, 97-144.
http://dx.doi.org/10.1007/BF03178905
[31]  Bradley, R.C. (2005) Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions, Probability Surveys, 2, 107-144.
[32]  Fryzlewicz, P. and Rao, S.S. (2011) Mixing Properties of ARCH and Time-Varying ARCH Processes. Bernoulli, 17, 320-346.
http://dx.doi.org/10.3150/10-BEJ270
[33]  Carrasco, M. and Chen, X. (2002) Mixing and Moment Properties of Various GARCH and Stochastic Volatility Models. Econometric Theory, 18, 17-39.
http://dx.doi.org/10.1017/S0266466602181023
[34]  Volkonskii, V.A. and Rozanov, Y.A. (1959) Some Limit Theorems for Random Functions I. Theory of Probability and Its Applications, 4, 178-197.
http://dx.doi.org/10.1137/1104015

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