Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
associated process; density deconvolution; nonstandard noise density
associated process; density deconvolution; nonstandard noise density
Summary:
We study the density deconvolution problem when the random variables of interest are an associated strictly stationary sequence and the random noises are i.i.d.\ with a nonstandard density. Based on a nonparametric strategy, we introduce an estimator depending on two parameters. This estimator is shown to be consistent with respect to the mean integrated squared error. Under additional regularity assumptions on the target function as well as on the density of noises, some error estimates are derived. Several numerical simulations are also conducted to illustrate the efficiency of our method.
References:
[1] Bagai, I., Rao, B. L. S. Prakasa: Kernel-type density and failure rate estimation for associated sequences. Ann. Inst. Stat. Math. 47 (1995), 253-266. DOI 10.1007/BF00773461 | MR 1345422 | Zbl 0833.62036
[2] Birkel, T.: On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 16 (1988), 1685-1698. DOI 10.1214/aop/1176991591 | MR 0958210 | Zbl 0658.60039
[3] Butucea, C., Tsybakov, A. B.: Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl. 52 (2008), 24-39. DOI 10.1137/S0040585X97982840 | MR 2354572 | Zbl 1141.62021
[4] Carrasco, M., Florens, J.-P.: A spectral method for deconvolving a density. Econom. Theory 27 (2011), 546-581. DOI 10.1017/S026646661000040X | MR 2806260 | Zbl 1218.62025
[5] Carroll, R. J., Hall, P.: Optimal rates of convergence for deconvolving a density. J. Am. Stat. Assoc. 83 (1988), 1184-1186. DOI 10.2307/2290153 | MR 0997599 | Zbl 0673.62033
[6] Chesneau, C.: On the adaptive wavelet deconvolution of a density for strong mixing sequences. J. Korean Stat. Soc. 41 (2012), 423-436. DOI 10.1016/j.jkss.2012.01.005 | MR 3255347 | Zbl 1296.62079
[7] Comte, F., Dedecker, J., Taupin, M.-L.: Adaptive density deconvolution with dependent inputs. Math. Methods Stat. 17 (2008), 87-112. DOI 10.3103/S1066530708020014 | MR 2429122 | Zbl 1282.62087
[8] Comte, F., Rozenholc, Y., Taupin, M.-L.: Penalized contrast estimator for adaptive density deconvolution. Can. J. Stat. 34 (2006), 431-452. DOI 10.1002/cjs.5550340305 | MR 2328553 | Zbl 1104.62033
[9] Dedecker, J., Prieur, C.: New dependence coefficients. Examples and applications to statistics. Probab. Theory Relat. Fields 132 (2005), 203-236. DOI 10.1007/s00440-004-0394-3 | MR 2199291 | Zbl 1061.62058
[10] Delaigle, A., Meister, A.: Nonparametric function estimation under Fourier-oscillating noise. Stat. Sin. 21 (2011), 1065-1092. DOI 10.5705/ss.2009.082 | MR 2827515 | Zbl 1232.62057
[11] Devroye, L.: Consistent deconvolution in density estimation. Can. J. Stat. 17 (1989), 235-239. DOI 10.2307/3314852 | MR 1033106 | Zbl 0679.62029
[12] Esary, J. D., Proschan, F., Walkup, D. W.: Association of random variables, with applications. Ann. Math. Stat. 38 (1967), 1466-1474. DOI 10.1214/aoms/1177698701 | MR 0217826 | Zbl 0183.21502
[13] Fan, J.: On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Stat. 19 (1991), 1257-1272. DOI 10.1214/aos/1176348248 | MR 1126324 | Zbl 0729.62033
[14] Fortuin, C. M., Kasteleyn, P. W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22 (1971), 89-103. DOI 10.1007/BF01651330 | MR 0309498 | Zbl 0346.06011
[15] Groeneboom, P., Jongbloed, G.: Density estimation in the uniform deconvolution model. Staat. Neerl. 57 (2003), 136-157. DOI 10.1111/1467-9574.00225 | MR 2035863 | Zbl 1090.62527
[16] Hall, P., Meister, A.: A ridge-parameter approach to deconvolution. Ann. Stat. 35 (2007), 1535-1558. DOI 10.1214/009053607000000028 | MR 2351096 | Zbl 1147.62031
[17] Lehmann, E. L.: Some concepts of dependence. Ann. Math. Stat. 37 (1966), 1137-1153. DOI 10.1214/aoms/1177699260 | MR 0202228 | Zbl 0146.40601
[18] Levin, B. Y.: Lectures on Entire Functions. Translations of Mathematical Monographs 150. AMS, Providence (1996). DOI 10.1090/mmono/150 | MR 1400006 | Zbl 0856.30001
[19] Liu, M. C., Taylor, R. L.: A consistent nonparametric density estimator for the deconvolution problem. Can. J. Stat. 17 (1989), 427-438. DOI 10.2307/3315482 | MR 1047309 | Zbl 0694.62017
[20] Masry, E.: Multivariate probability density deconvolution for stationary random processes. IEEE Trans. Inf. Theory 37 (1991), 1105-1115. DOI 10.1109/18.87002 | MR 1111811 | Zbl 0732.60045
[21] Masry, E.: Asymptotic normality for deconvolution estimators of multivariate densities of stationary processes. J. Multivariate Anal. 44 (1993), 47-68. DOI 10.1006/jmva.1993.1003 | MR 1208469 | Zbl 0783.62065
[22] Masry, E.: Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stochastic Processes Appl. 47 (1993), 53-74. DOI 10.1016/0304-4149(93)90094-K | MR 1232852 | Zbl 0797.62071
[23] Masry, E.: Deconvolving multivariate kernel density estimates from contaminated associated observations. IEEE Trans. Inf. Theory 49 (2003), 2941-2952. DOI 10.1109/TIT.2003.818415 | MR 2027570 | Zbl 1302.62085
[24] Meister, A.: Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions. Inverse Probl. 24 (2008), Article ID 015003, 14 pages. DOI 10.1088/0266-5611/24/1/015003 | MR 2384762 | Zbl 1143.65106
[25] Meister, A.: Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics 193. Springer, Berlin (2009). DOI 10.1007/978-3-540-87557-4 | MR 2768576 | Zbl 1178.62028
[26] Oliveira, P. E.: Asymptotics for Associated Random Variables. Springer, Berlin (2012). DOI 10.1007/978-3-642-25532-8 | MR 3013874 | Zbl 1249.62001
[27] Pan, J.: Tail dependence of random variables from ARCH and heavy-tailed bilinear models. Sci. China, Ser. A 45 (2002), 749-760. DOI 10.1360/02ys9082 | MR 1915886 | Zbl 1098.62549
[28] Pensky, M., Vidakovic, B.: Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Stat. 27 (1999), 2033-2053. DOI 10.1214/aos/1017939249 | MR 1765627 | Zbl 0962.62030
[29] Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987). MR 0924157 | Zbl 0925.00005
[30] Stefanski, L. A., Carroll, R. J.: Deconvoluting kernel density estimators. Statistics 21 (1990), 169-184. DOI 10.1080/02331889008802238 | MR 1054861 | Zbl 0697.62035
[31] Es, B. van, Spreij, P., Zanten, H. van: Nonparametric volatility density estimation. Bernoulli 9 (2003), 451-465. DOI 10.3150/bj/1065444813 | MR 1997492 | Zbl 1044.62037
[32] Zanten, H. van, Zareba, P.: A note on wavelet density deconvolution for weakly dependent data. Stat. Inference Stoch. Process. 11 (2008), 207-219. DOI 10.1007/s11203-007-9013-0 | MR 2403107 | Zbl 1204.62051
[33] Volkonskij, V. A., Rozanov, Y. A.: Some limit theorems for random functions. I. Theor. Probab. Appl. 4 (1959), 178-197. DOI 10.1137/1104015 | MR 0121856 | Zbl 0092.33502
Search
Partner of
