Abstract
In the convolution model Z i = X i + ε i , we give a model selection procedure to estimate the density of the unobserved variables (X i )1≤i≤n , when the sequence (X i ) i≥1 is strictly stationary but not necessarily independent. This procedure depends on whether the density of the ɛ i is supersmooth or ordinary smooth. The rates of convergence of the penalized contrast estimators are the same as in the independent framework, and are minimax over most regularity classes on ℝ. Our results apply to mixing sequences, but also to many other dependent sequences. When the errors are supersmooth, the condition on the dependence coefficients is the minimal condition of that type ensuring that the sequence (X i ) i≥1 is not a long-memory process.
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References
H. Berbee, Random Walks with Stationary Increments and Renewal Theory, in Mathematical Centre Tracts, Vol. 112 (Mathematisch Centrum, Amsterdam, 1979).
L. Birgé and P. Massart, “Minimum Contrast Estimators on Sieves: Exponential Bounds and Rates of Convergence”, Bernoulli 4, 329–375 (1998).
R. C. Bradley, Introduction to StrongMixing Conditions (Kendrick Press, Heber City, UT, 2007), Vol. 1.
C. Butucea, “Deconvolution of Supersmooth Densities with Smooth Noise”, Canadian J. Statist. 32, 181–192 (2004).
C. Butucea and A. B. Tsybakov, “Sharp Optimality for Density Deconvolution with Dominating Bias”, Teor. Veroyatn. Primen. 52, 111–128 (2007).
F. Comte, “Kernel Deconvolution of Stochastic VolatilityModels”, J. Time Ser. Anal. 25, 563–582 (2004).
F. Comte and V. Genon-Catalot, “Penalized Projection Estimator for Volatility Density”, Scand. J. Statist. 33, 875–893 (2006).
F. Comte and F. Merlevéde, “Adaptive Estimation of the Stationary Density of Discrete and Continuous Time Mixing Processes”, ESAIMProbab. Statist. (Electronic) 6, 211–238 (2002).
F. Comte, Y. Rozenholc, and M. L. Taupin, “Penalized Contrast Estimator for Adaptive Density Deconvolution”, Canadian J. Statist. 34, 431–452 (2006).
F. Comte, Y. Rozenholc, and M. L. Taupin, “Finite Sample Penalization in Adaptive Density Deconvolution”, J. Statist. Comput. Simul. 77, 977–1000 (2007).
J. Dedecker and C. Prieur, “NewDependenceCoefficients.Examples and Applications to Statistics”, Probab. Theory Rel. Fields 132, 203–236 (2005).
L. Devroye, Non-Uniform Random Variate Generation (Springer, New York etc., 1986).
P. Doukhan, Mixing: Properties and Examples, in Lecture Notes in Statistics, Vol. 85 (New York, Springer, 1994).
J. D. Esary, F. Proschan, and D. W. Walkup, “Association of Random Variables, with Applications”, Ann. Math. Statist. 38, 1466–1474 (1967).
J. Fan, “On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems”, Ann. Statist. 19, 1257–1272 (1991).
T. Klein and E. Rio, “Concentration around theMean forMaxima of Empirical Processes”, Ann. Probab. 33, 1060–1077 (2005).
C. Lacour, “Rates of Convergence for Nonparametric Deconvolution”, C. R. Math. Acad. Sci. Paris Sér. I Math. 342, 877–882 (2006).
E. Masry, “Strong Consistency and Rates for Deconvolution of Multivariate Densities of Stationary Processes”, Stochastic Processes Appl. 47, 53–74 (1993).
E. Masry, “Deconvolving Multivariate Kernel Density Estimates from Contaminated Associated Observations”, IEEE Trans. Inform. Theory 49, 2941–2952 (2003).
Y. Meyer, Ondelettes et opérateurs I: Ondelettes, in Actualités Mathéematiques (Hermann, Paris, 1990).
M. Pensky, “Estimation of a Smooth Density Function Using Meyer-Type Wavelets”, Statist. Decis. 17, 111–123 (1999).
M. Pensky and B. Vidakovic, “AdaptiveWavelet Estimator for Nonparametric Density Deconvolution”, Ann. Statist. 27, 2033–2053 (1999).
P. Rigollet, “Adaptive Density Estimation Using the Blockwise Stein Method”, Bernoulli 12, 351–370 (2006).
M. Rosenblatt, “A Central Limit Theorem and a Strong Mixing Condition”, Proc. Nat. Acad. Sci. USA 42, 43–47 (1956).
K. Tribouley and G. Viennet, “\( \mathbb{L}_p \) Adaptive Density Estimation in a β Mixing Framework”, Ann. Inst. H. Poincaré, Probab. Statist. 34, 179–208 (1998).
B. van Es, P. Spreij, and H. van Zanten, “Nonparametric Volatility Density Estimation”, Bernoulli 9, 451–465 (2003).
B. van Es, P. Spreij, and H. van Zanten, “Nonparametric Volatility Density Estimation for Discrete Time Models”, J. Nonparam. Statist. 17, 237–251 (2005).
G. Viennet, “Inequalities for Absolutely Regular Sequences: Application to Density Estimation”, Probab. Theory Rel. Fields 107, 467–492 (1997).
V. A. Volkonskii and Yu. A. Rozanov, “Some Limit Theorems for Random Functions. I”, Theory Probab. Appl. 4, 178–197 (1960).
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Comte, F., Dedecker, J. & Taupin, M.L. Adaptive density deconvolution with dependent inputs. Math. Meth. Stat. 17, 87–112 (2008). https://doi.org/10.3103/S1066530708020014
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DOI: https://doi.org/10.3103/S1066530708020014