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Nonparametric estimation of conditional expectation. (English) Zbl 1149.62078
Summary: Denote the integer lattice points in the N-dimensional Euclidean space by N and assume that (X i ,Y i ), i N , is a mixing random field. Estimators of the conditional expectation r(x)=E[Y i |X i =x] by nearest neighbor methods are established and investigated. The main analytical result of this study is that, under general mixing assumptions, the estimators considered are asymptotically normal. Many difficulties arise since points in higher dimensional space N2 cannot be linearly ordered. Our result applies to many situations where parametric methods cannot be adopted with confidence.
MSC:
62M40Statistics of random fields; image analysis
62G05Nonparametric estimation
62E20Asymptotic distribution theory in statistics
62G08Nonparametric regression
62M10Time series, auto-correlation, regression, etc. (statistics)
References:
[1]Boente, G.; Fraiman, R.: Consistency of a nonparametric estimate of a density function for dependent variables, J. multivariate anal. 25, 90-99 (1988) · Zbl 0664.62038 · doi:10.1016/0047-259X(88)90154-6
[2]Boente, G.; Fraiman, R.: Asymptotic distribution of robust estimators for nonparametric models from mixing processes, Ann. statist. 18, 891-906 (1990) · Zbl 0703.62025 · doi:10.1214/aos/1176347631
[3]Boente, G.; Fraiman, R.: Asymptotic distribution of smoothers based on local means and local medians under dependence, J. multivariate anal. 54, 77-90 (1995) · Zbl 0898.62043 · doi:10.1006/jmva.1995.1045
[4]Bosq, D.: Estimation et prévision nonparamétrique d’un processus stationnaire, C. R. Acad. sci. Paris sér. I math. 308, 453-456 (1989) · Zbl 0666.62092
[5]Bradley, R. C.: Introduction to strong mixing conditions, Introduction to strong mixing conditions 3 (2005)
[6]Collomb, G., 1980. Estimation de la regression par la méthode des k points les plus proches avec noyau: Quelques propiétés de convergence ponctuelle. In: Nonparametric Asymptotic Statistics, Lecture Notes in Mathematics, vol. 821. Springer, Berlin, pp. 159 – 175. · Zbl 0445.62055
[7]Györfi, L.; Härdle, W.; Sarda, P.; Vieu, P.: Nonparametric curve estimation from time series. Lecture notes in statistics, Nonparametric curve estimation from time series. Lecture notes in statistics 60 (1989) · Zbl 0697.62038
[8]Hallin, M.; Tran, L. T.: Kernel density estimation for linear processes: asymptotic normality and optimal bandwidth derivation, Ann. inst. Statist. math. 48, 429-449 (1996) · Zbl 0886.62042 · doi:10.1007/BF00050847
[9]Hallin, M.; Lu, Z.; Tran, L. T.: Local linear spatial regression, Ann. statist. 32, 2469-2500 (2004) · Zbl 1069.62075 · doi:10.1214/009053604000000850
[10]Ibragimov, I. A.; Linnik, Yu.V.: Independent and stationary sequence of random variables, (1971) · Zbl 0219.60027
[11]Ioannides, D. A.; Roussas, G. G.: Note on the uniform convergence of density estimates for mixing random variables, Statist. probab. Lett. 5, 279-285 (1987) · Zbl 0624.62039 · doi:10.1016/0167-7152(87)90105-2
[12]Lu, Z.: Asymptotic normality of kernel density estimators under dependence, Ann. inst. Statist. math. 53, 447-468 (2001) · Zbl 0989.62021 · doi:10.1023/A:1014652626073
[13]Lu, Z.; Chen, X.: Spatial nonparametric regression estimation: non-isotropic case, Acta math. Appl. sinica (English ser.) 18, 641-656 (2002) · Zbl 1019.62039 · doi:10.1007/s102550200067
[14]Mack, Y. P.: Local properties of k-NN regression estimates, SIAM J. Algebraic discrete methods 2, 311-323 (1981) · Zbl 0499.62037 · doi:10.1137/0602035
[15]Masry, E.: Probability density estimation from sampled data, IEEE trans. Inform. theory 29, 696-709 (1983) · Zbl 0521.62031 · doi:10.1109/TIT.1983.1056736
[16]Masry, E.: Recursive probability density estimation for weakly dependent processes, IEEE trans. Inform. theory 32, 254-267 (1986) · Zbl 0602.62028 · doi:10.1109/TIT.1986.1057163
[17]Masry, E.; Györfi, L.: Strong consistency and rates for recursive density estimators for stationary processes, J. multivariate anal. 22, 79-93 (1987) · Zbl 0619.62079 · doi:10.1016/0047-259X(87)90077-7
[18]Masry, E.; Tjøstheim, D.: Nonparametric estimation and identification of nonlinear ARCH time series: strong consistency and asymptotic normality, Econometric theory 11, 258-289 (1995)
[19]Nakhapetyan, B. S.: An approach to the proof of limit theorems for dependent random variables, Teor. veroyatnost. I primenen. 32, 589-594 (1987) · Zbl 0629.60045
[20]Robinson, P. M.: Nonparametric estimators for time series, J. time ser. Anal. 4, 185-207 (1983) · Zbl 0544.62082 · doi:10.1111/j.1467-9892.1983.tb00368.x
[21]Robinson, P. M.: Time series residuals with application to probability density estimation, J. time ser. Anal. 8, 329-344 (1987) · Zbl 0625.62071 · doi:10.1111/j.1467-9892.1987.tb00445.x
[22]Roussas, G. G.: Nonparametric estimation of the transition distribution function of a Markov process, Ann. math. Statist. 40, 1386-1400 (1969) · Zbl 0188.50501 · doi:10.1214/aoms/1177697510
[23]Roussas, G. G.: Nonparametric estimation in mixing sequences of random variables, Statist. plann. Inference 18, 135-149 (1988) · Zbl 0658.62048 · doi:10.1016/0378-3758(88)90001-8
[24]Tran, L. T.: The L1 convergence of kernel density estimates under dependence, Canad. J. Statist. 17, 197-208 (1989) · Zbl 0688.62030 · doi:10.2307/3314848
[25]Tran, L. T.; Yakowitz, S.: Nearest neighbor estimators for random fields, J. multivariate anal. 44, 23-46 (1993) · Zbl 0764.62076 · doi:10.1006/jmva.1993.1002
[26]Wu, W. B.; Mielniczuk, J.: Kernel density estimation for linear processes, Ann. statist. 30, 1441-1459 (2002) · Zbl 1015.62034 · doi:10.1214/aos/1035844982
[27]Yakowitz, S.: Nearest-neighbour methods for time series analysis, J. time ser. Anal. 8, 235-247 (1987) · Zbl 0615.62115 · doi:10.1111/j.1467-9892.1987.tb00435.x