×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.