A Note On The Adaptive Estimation Of A Quadratic Functional From Dependent Observations

Abstract

We investigate the estimation of the integral of the square of a multidimensional unknown function f under mild assumptions on the model allowing dependence on the observations. We develop an adaptive estimator based on a plug-in approach and wavelet projections. Taking the mean absolute error and assuming that f has a certain degree of smoothness, we prove that our estimator attains a sharp rate of convergence. Applications are given for the biased density model, the nonparametric regression model and a GARCH-type model under some mixing dependence conditions (alpha-mixing or beta-mixing). A simulation study considering nonparametric regression models with dependent observations illustrates the usefulness of the proposed estimator.

Keywords

• Quadratic functional estimation,Plug-in approach,Wavelets,Rates of convergence,Mixing dependence

References

1. Barbedor, P. (2006). Analyse en composantes ind´ependantes par ondelettes, th`ese Universit´e Paris VII, http://tel.archives-ouvertes.fr/tel-00119428.
2. Bickel, P. J. and Ritov, Y. (1988). Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhya Serie A, 50, 381-393.
3. Brunel, E., Comte, F. and Guilloux, A. (2009). Nonparametric density estimation in presence of bias and censoring. Test, 18, 1, 166-194.
4. Butucea, C. and Comte, F. (2009). Adaptive estimation of linear functionals in the convolution model and applications, Bernoulli, 15, 1, 69-98.
5. Butucea, C. and Meziani, K. (2011). Quadratic functional estimation in inverse problems, Statistical Methodology, 8, 1, 31-41.
6. Cai, T. and Low, M. (2005). Non-quadratic estimators of a quadratic functional. The Annals of Statistics, 33, 2930–2956.
7. Cai, T. and Low, M. (2006). Optimal adaptive estimation of a quadratic functional. The Annals of Statistics, 34, 2298–2325.
8. Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory, 18, 17-39.

9. Chesneau, C. (2011). Adaptive wavelet estimation of a biased density for strongly mixing sequences, International Journal of Mathematics and Mathematical Sciences, Volume 2011, Article ID 604150, 21 pages.
10. 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, 1-21.
11. Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1993). Wavelets on the interval and fast wavelet transforms. Applied and Computational Harmonic Analysis, 24, 1, 54–81.
12. Cox, D. (1969). Some sampling problems in technology. In New Developments in Survey Sampling (N. L. Johnson and H. Smith, Jr., eds.). Wiley, New York, 506-527.
13. Davydov, Y. (1970). The invariance principle for stationary processes. Theor. Probab. Appl., 15, 3, 498-509.
14. Delaigle, A. and Gijbels, I. (2002). Estimation of integrated squared density derivatives from a contaminated sample, Journal of the Royal Statistical Society, B, 64, 869-886.
15. DeVore, R. and Popov, V. (1988). Interpolation of Besov spaces. Trans. Amer. Math. Soc., 305, 397-414.
16. Donoho, D. L. and Nussbaum, M. (1990). Minimax quadratic estimation of a quadratic functional. Journal of Complexity, 6, 290-323.
17. Doukhan, P. (1994). Mixing. Properties and Examples. Lecture Notes in Statistics 85. Springer Verlag, New York.
18. Efromovich, S. (2004). Density estimation for biased data. Annals of Statistics, 32, (3), 1137-1161.
19. Fryzlewicz, P. and Subba Rao, S. (2011).Mixing properties of ARCH and time-varying ARCH processes, Bernoulli, Volume 17, Number 1 (2011), 320-346.
20. Gayraud, G. and Tribouley, K. (1999). Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law. Statistics and Probability Letters, 44, 109-122.
21. Gin´e, E. and Nickl, R. (2008). A Simple Adaptive Estimator of the Integrated Square of a Density. Bernoulli, 14, 47-61.
22. H¨ardle, W. (1990). Applied Nonparametric Regression, Cambridge University Press.
23. H¨ardle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelet, Approximation and Statistical Applications, Lectures Notes in Statistics New York 129, Springer Verlag.
24. Hosseinioun, N., Doosti, H. and Niroumand, H.A. (2009). Wavelet-based estimators of the integrated squared density derivatives for mixing sequences. Pakistan Journal of Statistics, 25(3), 341-350.
25. Johnstone, I. M. (2001a). Chi-square oracle inequalities. In State of the art in probability and statistics (Leiden, 1999), Lecture Notes - Monograph Series, volume 36, 399-418. Institute of Mathematical Statistics, Beachwood, OH.
26. Johnstone, I. M. (2001b). Thresholding for weighted chi-square. Statistica Sinica, 11, 691-704.
27. Kerkyacharian, G. and Picard, D. (1996). Estimating nonquadratic functionals of a density using haar wavelets. The Annals of Statistics, 24 (2), 485-507.
28. Laurent, B. (2005). Adaptive estimation of a quadratic functional of a density by model selection. ESAIM PS, 9, 1-18.
29. Liang, H. (2011). Asymptotic normality of wavelet estimator in heteroscedastic model with α-mixing error. J Syst Sci Complex, 24, (4), 725-737.
30. Mallat, S. (2009). A wavelet tour of signal processing. Elsevier/ Academic Press, Amsterdam, third edition. The sparse way, With contributions from Gabriel Peyr´e.
31. Marron, J.S., Adak, S., Johnstone, I.M., Neumann, M.H. and Patil, P. (1998). Exact risk analysis of wavelet regression. J. Comput. Graph. Statist, 7, 278-309.
32. Meyer, Y. (1992). Wavelets and Operators. Cambridge University Press, Cambridge.
33. Petsa, A. and Sapatinas, T. (2010). Adaptive quadratic functional estimation of a weighted density by model selection. Statistics, 44, 571-585.
34. Prakasa Rao. B.L.S. (1999). Estimation of the integrated squared density derivatives by wavelets. Bull. Inform. Cyb., 31, 47-65.
35. Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Ac. Sc. USA, 42, 43-47.
36. Shao, Q.-M. (1995). Maximal inequality for partial sums of ρ-mixing sequences. Ann. Probab., 23, 948- 965.
37. Tsybakov, A.B. (2004). Introduction `a l’estimation non-param´etrique, Springer.
38. Vidakovic, B. (1999). Statistical Modeling by Wavelets. John Wiley & Sons, Inc., New York, 384 pp.
39. Viennet, G. (1997). Inequalities for absolutely regular processes: application to density estimation. Probab. Theory Related Fields, 107, 467-492.
40. White, H. and Domowitz, I. (1984). Nonlinear Regression with Dependent Observations. Econometrica, 52, 143-162.