## Electronic Journal of Statistics

### Adaptive procedure for Fourier estimators: application to deconvolution and decompounding

#### Abstract

The purpose of this paper is twofold. First, introduce a new adaptive procedure to select the optimal – up to a logarithmic factor – cutoff parameter for Fourier density estimators. Two inverse problems are considered: deconvolution and decompounding. Deconvolution is a typical inverse problem, for which our procedure is numerically simple and stable, a comparison is performed with penalized techniques. Moreover, the procedure and the proof of oracle bounds do not rely on any knowledge on the noise term. Second, for decompounding, i.e. non-parametric estimation of the jump density of a compound Poisson process from the observation of $n$ increments at timestep $\Delta$, build an unified adaptive estimator which is optimal – up to a logarithmic factor – regardless the behavior of $\Delta$.

#### Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 3424-3452.

Dates
First available in Project Euclid: 25 September 2019

https://projecteuclid.org/euclid.ejs/1569398615

Digital Object Identifier
doi:10.1214/19-EJS1602

#### Citation

Duval, Céline; Kappus, Johanna. Adaptive procedure for Fourier estimators: application to deconvolution and decompounding. Electron. J. Statist. 13 (2019), no. 2, 3424--3452. doi:10.1214/19-EJS1602. https://projecteuclid.org/euclid.ejs/1569398615

#### References

• [1] A. Barron, L. Birgé, and P. Massart. Risk bounds for model selection via penalization., Probability theory and related fields, 113(3):301–413, 1999.
• [2] J.-P. Baudry, C. Maugis, and B. Michel. Slope heuristics: overview and implementation., Statistics and Computing, 22(2):455–470, 2012.
• [3] L. Birgé, P. Massart, et al. Minimum contrast estimators on sieves: exponential bounds and rates of convergence., Bernoulli, 4(3):329–375, 1998.
• [4] B. Buchmann and R. Grübel. Decompounding: an estimation problem for Poisson random sums., Ann. Statist., 31(4) :1054–1074, 2003.
• [5] C. Butucea. Deconvolution of supersmooth densities with smooth noise., Canadian Journal of Statistics, 32(2):181–192, 2004.
• [6] C. Butucea and A. B. Tsybakov. Sharp optimality in density deconvolution with dominating bias. I., Teor. Veroyatn. Primen., 52(1):111–128, 2007.
• [7] C. Butucea and A. B. Tsybakov. Sharp optimality in density deconvolution with dominating bias. II., Teor. Veroyatn. Primen., 52(2):336–349, 2007.
• [8] R. J. Carroll and P. Hall. Optimal rates of convergence for deconvolving a density., Journal of the American Statistical Association, 83(404) :1184–1186, 1988.
• [9] A. J. Coca. Efficient nonparametric inference for discretely observed compound poisson processes., Probability Theory and Related Fields, 170(1-2):475–523, 2018.
• [10] A. J. Coca. Adaptive nonparametric estimation for compound poisson processes robust to the discrete-observation scheme., 1803.09849, [math.ST].
• [11] F. Comte, C. Duval, and V. Genon-Catalot. Nonparametric density estimation in compound poisson processes using convolution power estimators., Metrika, 77(1):163–183, 2014.
• [12] F. Comte and C. Lacour. Data-driven density estimation in the presence of additive noise with unknown distribution., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(4):601–627, 2011.
• [13] F. Comte and C. Lacour. Anisotropic adaptive kernel deconvolution. In, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, volume 49, pages 569–609. Institut Henri Poincaré, 2013.
• [14] F. Comte, Y. Rozenholc, and M.-L. Taupin. Finite sample penalization in adaptive density deconvolution., J. Stat. Comput. Simul., 77(11-12):977 –1000, 2007.
• [15] R. Cont and A. De Larrard. Price dynamics in a markovian limit order market., SIAM Journal on Financial Mathematics, 4(1):1–25, 2013.
• [16] D. L. Donoho and I. M. Johnstone. Ideal spatial adaptation by wavelet shrinkage., Biometrika, pages 425–455, 1994.
• [17] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard. Wavelet shrinkage: asymptopia?, Journal of the Royal Statistical Society. Series B (Methodological), pages 301–369, 1995.
• [18] C. Duval. Density estimation for compound Poisson processes from discrete data., Stochastic Process. Appl., 123(11) :3963–3986, 2013.
• [19] C. Duval. When is it no longer possible to estimate a compound poisson process?, Electronic Journal of Statistics, 8(1):274–301, 2014.
• [20] C. Duval and J. Kappus. Nonparametric adaptive estimation for grouped data., Journal of Statistical Planning and Inference, 182:12–28, 2017.
• [21] P. Embrechts, C. Klüppelberg, and T. Mikosch., Modelling extremal events: for insurance and finance, volume 33. Springer Science & Business Media, 2013.
• [22] J. Fan. On the optimal rates of convergence for nonparametric deconvolution problems., The Annals of Statistics, pages 1257–1272, 1991.
• [23] A. Goldenshluger and O. Lepski. Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality., The Annals of Statistics, pages 1608–1632, 2011.
• [24] A. Goldenshluger and O. Lepski. On adaptive minimax density estimation on $\mathbbR^d$., Probability Theory and Related Fields, 159(3-4):479–543, 2014.
• [25] S. Gugushvili, F. Van der Meulen, and P. Spreij. Nonparametric bayesian inference for multidimensional compound Poisson processes., Modern Stochastics: Theory and Applications, 2(1):1–15, Mar 2015.
• [26] J. Kappus and G. Mabon. Adaptive density estimation in deconvolution problems with unknown error distribution., Electronic Journal of Statistics, 8(2) :2879–2904, 2014.
• [27] C. Lacour, P. Massart, and V. Rivoirard. Estimator selection: a new method with applications to kernel density estimation., Sankhya A, 79(2):298–335, 2017.
• [28] O. V. Lepski and T. Willer. Oracle inequalities and adaptive estimation in the convolution structure density model., Ann. Statist., 47(1):233–287, 02 2019.
• [29] M. Lerasle. Optimal model selection in density estimation. In, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, volume 48, pages 884–908. Institut Henri Poincaré, 2012.
• [30] K. Lounici and R. Nickl. Uniform risk bounds and confidence bands in wavelet deconvolution., Annals of Statistics, 39:201–231, 2011.
• [31] P. Massart., Concentration inequalities and model selection, volume 6. Springer, 2007.
• [32] M. H. Neumann. On the effect of estimating the error density in nonparametric deconvolution., J. Nonparametr. Statist., 7(4):307–330, 1997.
• [33] M. H. Neumann and M. Reiß. Nonparametric estimation for Lévy processes from low-frequency observations., Bernoulli, 15(1):223–248, 2009.
• [34] M. Pensky, B. Vidakovic, et al. Adaptive wavelet estimator for nonparametric density deconvolution., The Annals of Statistics, 27(6) :2033–2053, 1999.
• [35] G. Rebelles. Structural adaptive deconvolution under $l_p$-losses., Mathematical Methods of Statistics, 25(1):26–53, 2016.
• [36] P. Reynaud-Bouret, V. Rivoirard, and C. Tuleau-Malot. Adaptive density estimation: a curse of support?, Journal of Statistical Planning and Inference, 141(1):115–139, 2011.
• [37] L. A. Stefanski. Rates of convergence of some estimators in a class of deconvolution problems., Statistics & Probability Letters, 9(3):229–235, 1990.
• [38] B. van Es, S. Gugushvili, and P. Spreij. A kernel type nonparametric density estimator for decompounding., Bernoulli, 13(3):672–694, 2007.