Estimation of convolution in the model with noise

References

• Ahmad, I.A., and Fan, Y. (2001), ‘Optimal Bandwidth for Kernel Density Estimators of Functions of Observations’, Statistics & Probability Letters, 51, 245–251. doi: 10.1016/S0167-7152(00)00136-X
   Web of Science ®Google Scholar
• Ahmad, I.A., and Mugdadi, A.R. (2003), ‘Analysis of Kernel Density Estimation of Functions of Random Variables’, Journal of Nonparametric Statistics, 15, 579–605. doi: 10.1080/10485250310001605441
   Web of Science ®Google Scholar
• Baudry, J.-P., Maugis, C., and Michel, B. (2012), ‘Slope Heuristics: Overview and Implementation’, Statistics and Computing, 22, 455–470. doi: 10.1007/s11222-011-9236-1
   Web of Science ®Google Scholar
• Bonhomme, S., and Robin, J.-M. (2010), ‘Generalized Non-parametric Deconvolution with an Application to Earnings Dynamics’, The Review of Economic Studies, 77, 491–533. doi: 10.1111/j.1467-937X.2009.00577.x
   Web of Science ®Google Scholar
• Butucea, C. (2004), ‘Deconvolution of Supersmooth Densities with Smooth Noise’, The Canadian Journal of Statistics, 32, 181–192. doi: 10.2307/3315941
   Web of Science ®Google Scholar
• Butucea, C., and Tsybakov, A.B. (2008), ‘Sharp Optimality in Density Deconvolution with Dominating Bias. II’, Theory Probability and its Application, 52, 237–249. doi: 10.1137/S0040585X97982992
   Web of Science ®Google Scholar
• Caroll, R.J., and Hall, P. (1988), ‘Optimal Rates of Convergence for Deconvolving a Density’, Journal of American Statistical Association, 83, 1184–1186. doi: 10.1080/01621459.1988.10478718
   Web of Science ®Google Scholar
• Chesneau, C., Comte, F., and Navarro, F. (2013), ‘Fast Nonparametric Estimation for Convolutions of Densities’, The Canadian Journal of Statistics, 41, 617–636. doi: 10.1002/cjs.11191
   Web of Science ®Google Scholar
• Chesneau, C., and Navarro, F. (2014), ‘On a Plug-in Wavelet Estimator for Convolutions of Densities’, Journal of Statistical Theory and Practice, 8, 653–673. doi: 10.1080/15598608.2013.824824
   Google Scholar
• Comte, F., and Lacour, C. (2011), ‘Data-driven Density Estimation in the Presence of Additive Noise with Unknown Distribution’, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73, 601–627. doi: 10.1111/j.1467-9868.2011.00775.x
   Web of Science ®Google Scholar
• Comte, F., Rozenholc, Y., and Taupin, M.-L. (2006), ‘Penalized Contrast Estimator for Density Deconvolution’, The Canadian Journal of Statistics, 34, 431–452. doi: 10.1002/cjs.5550340305
   Web of Science ®Google Scholar
• Delaigle, A., and Gijbels, I. (2006), ‘Estimation of Boundary and Discontinuity Points in Deconvolution Problems’, Statistica Sinica, 16, 773–788.
   Web of Science ®Google Scholar
• Delaigle, A., Hall, P., and Meister, A. (2008), ‘On Deconvolution with Repeated Measurements’, Annals of Statistics, 36, 665–685. doi: 10.1214/009053607000000884
   Web of Science ®Google Scholar
• Devroye, L. (1989), ‘Consistent Deconvolution in Density Estimation’, The Canadian Journal of Statistics, 17, 235–239. doi: 10.2307/3314852
   Web of Science ®Google Scholar
• Du, J., and Schick, A. (2007), ‘Root-n Consistency and Functional Central Limit Theorems for Estimators of Derivatives of Convolutions of Densities’, International Journal of Statistics & Management Systems, 2, 67–87.
   Google Scholar
• Fan, J. (1991), ‘On the Optimal Rates of Convergence for Nonparametric Deconvolution Problem’, Annals of Statistics, 19, 1257–1272. doi: 10.1214/aos/1176348248
   Web of Science ®Google Scholar
• Fan, J., and Koo, J.Y. (2002), ‘Wavelet Deconvolution’, IEEE Transactions on Information Theory, 48, 734–747. doi: 10.1109/18.986021
   Web of Science ®Google Scholar
• Frees, E. (1994), ‘Estimating Densities of Functions of Observations’, Journal of the American Statistical Association, 89, 17–525. doi: 10.1080/01621459.1994.10476776
   Google Scholar
• Giné, E., and Mason, D.M. (2007), ‘On Local U-statistic Processes and the Estimation of Densities of Functions of Several Sample Variables’, Annals of Statistics, 35, 1105–1145. doi: 10.1214/009053607000000154
   Web of Science ®Google Scholar
• Johannes, J. (2009), ‘Deconvolution with Unknown Error Distribution’, Annals of Statistics, 37, 2301–2323. doi: 10.1214/08-AOS652
   Web of Science ®Google Scholar
• Johannes, J, and Schwarz, M. (2013), ‘Adaptive Gaussian Inverse Regression with Partially Unknown Operator’, Communications in Statistics – Theory and Methods, 42, 1343–1362. doi: 10.1080/03610926.2012.731548
   Web of Science ®Google Scholar
• Kappus, J. (2014), ‘Adaptive Nonparametric Estimation for Lévy Processes Observed at Low Frequency’, Stochastic Process and their Applications, 124, 730–758. doi: 10.1016/j.spa.2013.08.010
   Web of Science ®Google Scholar
• Kappus, J., and Mabon, G. (2014), ‘Adaptive Density Estimation in Deconvolution Problems with Unknown Error Distribution’, Electronic Journal of Statistics, 8, 2879–2904. doi: 10.1214/14-EJS976
   Google Scholar
• Lacour, C. (2006), ‘Rates of Convergence for Nonparametric Deconvolution’, Comptes Rendus Mathematique. Academie des Sciences. Paris, 342, 877–882. doi: 10.1016/j.crma.2006.04.006
   Web of Science ®Google Scholar
• Mabon, G. (2015), ‘Adaptive Estimation of Marginal Random-effects Densities in Linear Mixed-effects Model’, HAL Id : hal-00958905, version 4. To appear in Mathematical Methods of Statistics.
   Google Scholar
• Marron, J.S., and Wand, M.P. (1992), ‘Exact Mean Integrated Squared Error’, Annals of Statistics, 20, 712–736. doi: 10.1214/aos/1176348653
   Web of Science ®Google Scholar
• Meister, A., Deconvolution Problems in Nonparametric Statistics. Vol. 193, Lecture Notes in Statistics, Berlin, Springer, 2009.
   Google Scholar
• Neumann, M.H. (1997), ‘On the Effect of Estimating the Error Density in Nonparametric Deconvolution’, Journal of Nonparametric Statistics, 7, 307–330. doi: 10.1080/10485259708832708
   Google Scholar
• Neumann, M.H. (2007), ‘Deconvolution from Panel Data with Unknown Error Distribution’, Journal of Multivariate Analysis, 98, 1955–1968. doi: 10.1016/j.jmva.2006.09.012
   Web of Science ®Google Scholar
• Neumann, M.H., and Reiss, M. (2009), ‘Nonparametric Estimation for Levy Processes from Low-frequency Observations’, Bernoulli, 15, 223–248. doi: 10.3150/08-BEJ148
   Web of Science ®Google Scholar
• Nickl, R. (2007), ‘Donsker-type Theorems for Nonparametric Maximum Likelihood Estimators’, Probability Theory and Related Fields, 138, 411–449. doi: 10.1007/s00440-006-0031-4
   Web of Science ®Google Scholar
• Nickl, R. (2009), ‘On Convergence and Convolutions of Random Signed Measures’, Journal of Theoretical Probability, 22, 38–56. doi: 10.1007/s10959-008-0177-3
   Web of Science ®Google Scholar
• Pensky, M., and Vidakovic, B. (1999), ‘Adaptive Wavelet Estimator for Nonparametric Density Deconvolution’, Annals of Statistics, 27, 2033–2053. doi: 10.1214/aos/1017939249
   Web of Science ®Google Scholar
• Prakasa Rao, B.L.S. (2004), ‘Moment Inequalities for Supremum of Empirical Processes of U-statistic Structure and Application to Density Estimation’, Journal of Iranian Statistical Society, 3, 59–68.
   Google Scholar
• Rosenthal, H.P. (1970), ‘On the Subspaces of () Spanned by Sequences of Independent Random Variables’, Israel Journal of Mathematics, 8, 273–303. doi: 10.1007/BF02771562
   Web of Science ®Google Scholar
• Saavedra, A., and Cao, R. (2000), ‘On the Estimation of the Marginal Density of a Moving Average Process’, Canadian Journal of Statistics, 28, 799–815. doi: 10.2307/3315917
   Web of Science ®Google Scholar
• Schick, A., and Wefelmeyer, W. (2004), ‘Root n Consistent Density Estimators for Sums of Independent Random Variables’, Journal of Nonparametric Statistics, 16, 925–935. doi: 10.1080/10485250410001713990
   Web of Science ®Google Scholar
• Schick, A., and Wefelmeyer, W. (2007), ‘Root n Consistent Density Estimators of Convolutions in Weighted -norms’, Journal of Statistical Planning Inference, 137, 1765–1774. doi: 10.1016/j.jspi.2006.06.041
   Web of Science ®Google Scholar
• Shiryaev, A.N. (1996), Probability. Translated from the first (1980) Russian edition by R. P. Boas. Second edition. Graduate Texts in Mathematics, Vol. 95. Springer, New York.
   Google Scholar
• Silverman, B.W. (1986), Density Estimation: for Statistics and Data Analysis, London: Chapman & Hall.
   Google Scholar