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A Journal of Theoretical and Applied Statistics
Volume 52, 2018 - Issue 6
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Original Articles

Non-parametric regression for spatially dependent data with wavelets

Pages 1270-1308 | Received 29 Jun 2017, Accepted 04 Jul 2018, Published online: 07 Aug 2018

References

  • Györfi L, Kohler M, Krzyżak A, et al. A distribution-free theory of nonparametric regression. New York, Heidelberg: Springer Berlin; 2002.
  • Györfi L, Härdle W, Sarda P, et al. Nonparametric curve estimation from time series. Vol. 60. Berlin: Springer; 1989.
  • Härdle W. Applied nonparametric regression. Cambridge: Cambridge University Press; 1990.
  • Grenander U. Abstract inference. New York: John Wiley & Sons; 1981.
  • Hornik K. Approximation capabilities of multilayer feedforward networks. Neural Netw. 1991;4(2):251–257. doi: 10.1016/0893-6080(91)90009-T
  • Barron AR. Approximation and estimation bounds for artificial neural networks. Mach Learn. 1994;14(1):115–133.
  • McCaffrey DF, Gallant AR. Convergence rates for single hidden layer feedforward networks. Neural Netw. 1994;7(1):147–158. doi: 10.1016/0893-6080(94)90063-9
  • Franke J, Diagne M. Estimating market risk with neural networks. Statist Decis. 2006;24:1001–1021.
  • Kirch C, Tadjuidje Kamgaing J. A uniform central limit theorem for neural network-based autoregressive processes with applications to change-point analysis. Statistics. 2014;48(6):1187–1201. doi: 10.1080/02331888.2013.872646
  • Fan J, Gijbels I. Local polynomial modelling and its applications. London: Chapman and Hall; 1995.
  • Härdle W, Kerkyacharian G, Picard D, et al. Wavelets, approximation, and statistical applications. New York: Springer; 2012. (Lecture Notes in Statistics).
  • Donoho DL, Johnstone IM, Kerkyacharian G, et al. Density estimation by wavelet thresholding. Ann Statist. 1996;24:508–539. doi: 10.1214/aos/1032894451
  • Donoho DL, Johnstone IM. Minimax estimation via wavelet shrinkage. Ann Statist. 1998;26(3):879–921. doi: 10.1214/aos/1024691081
  • Cai TT. Adaptive wavelet estimation: a block thresholding and oracle inequality approach. Ann Statist. 1999;27:898–924. doi: 10.1214/aos/1018031262
  • Kerkyacharian G, Picard D. Regression in random design and warped wavelets. Bernoulli. 2004;10(6):1053–1105. doi: 10.3150/bj/1106314850
  • Kulik R, Raimondo M. Wavelet regression in random design with heteroscedastic dependent errors. Ann Statist. 2009;37(6A):3396–3430. doi: 10.1214/09-AOS684
  • Brown LD, Cai TT, Zhou HH. Nonparametric regression in exponential families. Ann Statist. 2010;38(4):2005–2046. doi: 10.1214/09-AOS762
  • Cressie N. Statistics for spatial data. New York: J. Wiley; 1993. (Wiley series in probability and mathematical statistics: Applied probability and statistics).
  • Kindermann R, Snell L. Markov random fields and their applications. Providence (RI): The American Mathematical Society; 1980.
  • Baraud Y, Comte F, Viennet G. Adaptive estimation in autoregression or β-mixing regression via model selection. Ann Statist. 2001;29:839–875. doi: 10.1214/aos/1009210692
  • Guessoum Z, Sa”ıd EO. Kernel regression uniform rate estimation for censored data under α-mixing condition. Electron J Stat. 2010;4:117–132. doi: 10.1214/08-EJS195
  • Koul HL. Behavior of robust estimators in the regression model with dependent errors. Ann Statist. 1977;5:681–699. doi: 10.1214/aos/1176343892
  • Li L, Xiao Y. Wavelet-based estimation of regression function with strong mixing errors under fixed design. Comm Statist Theory Methods. 2017;46(10):4824–4842. doi: 10.1080/03610926.2015.1089288
  • Roussas GG, Tran LT. Asymptotic normality of the recursive kernel regression estimate under dependence conditions. Ann Statist. 1992;20:98–120. doi: 10.1214/aos/1176348514
  • Yahia D, Benatia F. Nonlinear wavelet regression function estimator for censored dependent data. Afr Stat. 2012;7(1):391–411.
  • Carbon M, Francq C, Tran LT. Kernel regression estimation for random fields. J Stat Plan Inference. 2007;137(3):778–798. doi: 10.1016/j.jspi.2006.06.008
  • Hallin M, Lu Z, Tran LT. Local linear spatial regression. Ann Statist. 2004;32(6):2469–2500. doi: 10.1214/009053604000000850
  • Hall P, Patil P. Formulae for mean integrated squared error of nonlinear wavelet-based density estimators. Ann Statist. 1995;23:905–928. doi: 10.1214/aos/1176324628
  • Li L. Nonparametric regression on random fields with random design using wavelet method. Stat Inference Stoch Process. 2016;19(1):51–69. doi: 10.1007/s11203-015-9119-8
  • Kaiser MS, Lahiri SN, Nordman DJ. Goodness of fit tests for a class of markov random field models. Ann Statist. 2012;40:104–130. doi: 10.1214/11-AOS948
  • Rosenblatt M. A central limit theorem and a strong mixing condition. Proc Natl Acad Sci. 1956;42(1):43–47. doi: 10.1073/pnas.42.1.43
  • Doukhan P. Mixing: properties and examples. Universite Paris-sud, Departement de mathematique; 1991.
  • Kolmogorov AN, Rozanov YA. On strong mixing conditions for stationary gaussian processes. Theory Probab Appl. 1960;5(2):204–208. doi: 10.1137/1105018
  • Bradley RC. Basic properties of strong mixing conditions. A survey and some open questions. Probab surveys. 2005;2(2):107–144. doi: 10.1214/154957805100000104
  • Davydov YA. Mixing conditions for Markov chains. Teoriya Veroyatnostei i ee Primeneniya. 1973;18(2):321–338.
  • Withers CS. Conditions for linear processes to be strong-mixing. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 1981;57(4):477–480. doi: 10.1007/BF01025869
  • Chen X, Christensen T. Optimal uniform convergence rates for sieve nonparametric instrumental variables regression. Cowles Foundation Discussion Paper No. 1923. 2013.
  • Bradley RC. A caution on mixing conditions for random fields. Stat Probab Lett. 1989;8(5):489–491. doi: 10.1016/0167-7152(89)90032-1
  • Haussler D. Decision theoretic generalizations of the pac model for neural net and other learning applications. Inform Comput. 1992;100(1):78–150. doi: 10.1016/0890-5401(92)90010-D
  • Kohler M. Nonlinear orthogonal series estimates for random design regression. J Stat Plan Inference. 2003;115(2):491–520. doi: 10.1016/S0378-3758(02)00158-1
  • Stone CJ. Optimal global rates of convergence for nonparametric regression. Ann Statist. 1982;10:1040–1053. doi: 10.1214/aos/1176345969
  • Daubechies I. Ten lectures on wavelets. Philadelphia (PA): SIAM; 1992.
  • Meyer Y. Wavelets and operators. Vol. 1. Cambridge: Cambridge university press; 1995.
  • Strichartz RS. Construction of orthonormal wavelets. In: Benedetto JJ, editor. Wavelets: mathematics and applications. New York: CRC press; 1993. p. 23–50.
  • Krebs JTN. Orthogonal series estimates on strong spatial mixing data. J Stat Plan Inference. 2018;193:15–41. doi: 10.1016/j.jspi.2017.07.005
  • Berbee HC. Random walks with stationary increments and renewal theory. MC Tracts. 1979.
  • Carbon M, Tran LT, Wu B. Kernel density estimation for random fields (density estimation for random fields). Stat Probab Lett. 1997;36(2):115–125. doi: 10.1016/S0167-7152(97)00054-0
  • Tran LT. Kernel density estimation on random fields. J Multivar Anal. 1990;34(1):37–53. doi: 10.1016/0047-259X(90)90059-Q
  • Krebs JTN. A large deviation inequality for β-mixing time series and its applications to the functional kernel regression model. Stat Probab Lett. 2018;133:50–58. doi: 10.1016/j.spl.2017.09.013
  • Davydov YA. Convergence of distributions generated by stationary stochastic processes. Theory Probab Appl. 1968;13(4):691–696. doi: 10.1137/1113086
  • Merlevède F, Peligrad M, Rio E, et al. Bernstein inequality and moderate deviations under strong mixing conditions. In: High dimensional probability V: the Luminy volume. Beachwood (OH): Institute of Mathematical Statistics; 2009. p. 273–292.

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