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Dependence and the dimensionality reduction principle. (English) Zbl 1056.62054
Summary: C. Stone’s dimensionality reduction principle [Ann. Stat. 13, 689–705 (1985; Zbl 0605.62065); ibid. 10, 1040–1053 (1982; Zbl 0511.62048)] has been confirmed on several occasions for independent observations. When dependence is expressed with $$\varphi$$-mixing, a minimum distance estimate $$\widehat\theta_n$$ is proposed for a smooth projection pursuit regression-type function $$\theta\in\Theta$$, that is either additive or multiplicative, in the presence of or without interactions. Upper bounds on the $$L_1$$-risk and the $$L_1$$-error of $$\widehat\theta_n$$ are obtained, under restrictions on the order of decay of the mixing coefficient. The bounds show explicitly the additive effect of $$\varphi$$-mixing on the error, and confirm the dimensionality reduction principle.
##### MSC:
 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference
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##### References:
 [1] Beran, R. J. (1977). Minimum Hellinger distance estimates for parametric models,Annals of Statistics,5, 445–463. · Zbl 0381.62028 [2] Chaudhuri, P. (1991a). Global nonparametric estimation of conditional quantile functions and their derivatives,Journal of Multivariate Analysis,39, 246–269. · Zbl 0739.62028 [3] Chaudhuri, P. (1991b). Nonparametric estimates of regression quantiles and their local Bahadur representation,Annals of Statistics,19, 760–777. · Zbl 0728.62042 [4] Chen, H. (1991). Estimation of a projection-pursuit regression model,Annals of Statistics,19, 142–157. · Zbl 0736.62055 [5] Devroye, L. P. (1987).A Course in Density Estimation, Birkhauser, Boston. · Zbl 0617.62043 [6] Donoho, D. L. and Liu, R.C. (1988a). The ”automatic” robustness of minimum distance functionals,Annals of Statistics,16, 552–586. · Zbl 0684.62030 [7] Donoho, D. L. and Liu, R. C. (1988b). Pathologies of some minimum distance estimators,Annals of Statistics,16, 587–608. · Zbl 0684.62029 [8] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia?,Journal of the Royal Statistical Society, Series B,57, 301–369. · Zbl 0827.62035 [9] Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression,Journal of the American Statistical Association,76, 817–823. [10] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables,Journal of the American Statistical Association,58, 13–31. · Zbl 0127.10602 [11] Huber, P. J. (1985). Projection pursuit,Annals of Statistics,13, 435–475. · Zbl 0595.62059 [12] Ibragimov, I. A. and Khas’minskii R. Z. (1981).Statistical Estimation: Asymptotic Theory, Springer, New York. · Zbl 0467.62026 [13] Kolmogorov, A. N. and Tikhomirov, V. M. (1959). {$$\epsilon$$}-entropy and {$$\epsilon$$}-capacity of sets in function spaces,Uspekhi Matematicheskikh Nauk,14(2), 3–86 (in Russian) (1961).American Mathematical Society Translations (2),17, 277–364). · Zbl 0090.33503 [14] Le Cam, L. M. (1973). Convergence of estimates under under dimensionality restrictions,Annals of Statistics,1, 38–53. · Zbl 0255.62006 [15] Le Cam, L. M. (1986).Asymptotic Methods in Statistical Decision Theory, Springer, New York. · Zbl 0605.62002 [16] Le Cam, L. M. and Yang, G. L. (1990).Asymptotics in Statistics: Some Basic Concepts, Springer, New York. · Zbl 0719.62003 [17] Millar, P. W. (1981). Robust estimation via minimum distance methods,Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete,55, 73–89. · Zbl 0461.62036 [18] Nicoleris, T. and Yatracos, Y. G. (1997). Rates of convergence of estimates, Kolmogorov’s entropy and the dimensionality reduction principle in regression,Annals of Statistics,25, 2493–2511. · Zbl 0909.62063 [19] Roussas, G. G. and Ioannides, D. (1987). Moment inequalities for mixing sequences of random variables,Stochastic Analysis and Applications,5, 61–120. · Zbl 0619.60022 [20] Roussas, G. G. and Yatracos, Y. G. (1996). Minimum distance regression-type estimates with rates under weak dependence,Annals of the Institute of Statistical Mathematics,48 267–281. · Zbl 0859.62080 [21] Stone, C. J. (1982). Optimal global rates of convergence in nonparametric regression,Annals of Statistics,10, 1040–1053. · Zbl 0511.62048 [22] Stone, C. J. (1985). Additive regression and other nonparametric models,Annals of Statistics,13, 689–705. · Zbl 0605.62065 [23] Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models,Annals of Statistics,14, 590–606. · Zbl 0603.62050 [24] Stone, C. J. (1994). The use of polynomial splines and their tensor product in multivariate function estimation,Annals of Statistics,22, 118–184. · Zbl 0827.62038 [25] Tran, L. T. (1993). Nonparametric function estimation for time series by local average estimators,Annals of Statistics,21, 1040–1057. · Zbl 0790.62037 [26] Truong, Y. K. and Stone, C. J. (1992). Nonparametric function estimation involving time series,Annals of Statistics,20, 77–97. · Zbl 0764.62038 [27] Wolfowitz, J. (1957). The minimum distance method,Annals of Mathematical Statistics,28, 75–88. · Zbl 0086.35403 [28] Yatracos, Y. G. (1985). Rates of convergence of minimum distance estimators and Kolmogorov’s entropy,Annals of Statistics,13, 768–774. · Zbl 0576.62057 [29] Yatracos, Y. G. (1988). A lower bound on the error in nonparametric regression type problems,Annals of Statistics,16, 1180–1187. · Zbl 0651.62028 [30] Yatracos, Y. G. (1989a). A regression type problem,Annals of Statistics,17, 1597–1607. · Zbl 0694.62018 [31] Yatracos, Y. G. (1989b). On the estimation of the derivatives of a function via the derivatives of an estimate,Journal of Multivariate Analysis,28, 172–175. · Zbl 0665.62041 [32] Yatracos, Y. G. (1992).L 1-optimal estimates for a regression type function inR d Journal of Multivariate Analysis,40, 213–220. · Zbl 0744.62064
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