×

Weak and strong uniform consistency of kernel regression estimates. (English) Zbl 0495.62046


MSC:

62G05 Nonparametric estimation
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bertrand-Retali, M., Convergence uniforme stochastique d’un estimator d’une densité de probabilité dans ℝ^s, C.R. Acad. Sci. Paris, 278, 1449-1452 (1974)
[2] Bhattacharya, P. K., An invariance principle in regression analysis, Ann. Statist., 4, 621-624 (1976) · Zbl 0331.62016
[3] Bickel, P. J.; Rosenblatt, M., On some global measures of the deviations of density function estimates, Ann. Statist., 1, 1071-1095 (1973) · Zbl 0275.62033
[4] Bleuez, J.; Bosq, D., Conditions nécessaires et suffisantes de convergence pour une class d’estimateurs de la densité, C.R. Acad. Sci. Paris, 282, 636-666 (1976) · Zbl 0322.62043
[5] Bosq, D.: Contribution à la théorie de l’estimation fonctionnelle. Publications de l’Institut de Statistique de l’Université de Paris, Vol. 19, fasc. 2 et 3 (1970) · Zbl 0236.62024
[6] Cheng, K. F.; Taylor, R. L., On the uniform complete convergence of estimates for multivariate density functions and regression curves, Ann. Inst. Statist. Math., 32, 187-199 (1980) · Zbl 0459.62039
[7] Collomb, G., Estimation non-paramétrique de la régression par la methode du noyau, Thèse (1976), Toulouse: Université Paul Sabatier, Toulouse
[8] Collomb, G., Quelques propriétés de la méthode du noyau pour l’estimation non-paramétrique de la régression en un point fixé, C.R. Acad. Sci. Paris, 285, 289-292 (1977) · Zbl 0375.62042
[9] Collomb, G., Conditions nécessaires et suffisantes de convergence uniforme d’un estimateur de la régression, estimation des dérivées de la regression, C.R. Acad. Sci. Paris, 288, 161-164 (1979) · Zbl 0397.62042
[10] Collomb, G., Estimation non paramétrique de la régression: revue bibliographique, ISR, 49, 75-93 (1981) · Zbl 0471.62039
[11] Csörgő, M.; Révész, P., A new method to prove Strassen type laws of invariance principle II, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 31, 261-269 (1975) · Zbl 0283.60024
[12] Deheuvels, P., Conditions nécessaires et suffisantes de convergence ponctuelle presque sûre et uniforme presque sûre des estimateurs de la densité, C.R. Acad. Sci. Paris, 278, 1217-1220 (1974) · Zbl 0281.62041
[13] Devroye, L. P., The uniform convergence of the Nadaraya-Watson regression function estimate, Canad. J. Statist., 6, 179-191 (1979) · Zbl 0405.62033
[14] Garsia, A.M.: Continuity properties of Gaussian processes with multidimensional time parameter. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. 2, 369-374. Univ. of California Press (1970)
[15] Geffroy, J.: Étude de la convergence du régressogramme. Séminarie de Statistique Mathématique de l’Institut de Mathématique de l’Université de Paris, année 1974-1975 (1975)
[16] Hall, P., Laws of the iterated logarithm for nonparametric density estimators, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 50, 47-61 (1981) · Zbl 0443.62027
[17] Johnston, G.J.: Smooth nonparametric regression analysis. Ph.D. dissertation, U. of No. Carolina at Chapel Hill (1979)
[18] Komlós, J.; Major, P.; Tusnády, G., An approximation of partial sums of independent random variables, and the sample distribution function I, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 32, 111-131 (1975) · Zbl 0308.60029
[19] Konakov, V. D., On a global measure of deviation for an estimate of the regression line, Theor. Probab. Appl., 22, No. 4, 858-868 (1977) · Zbl 0391.62030
[20] Mack, Y. P., Local properties of k—NN regression estimates. SIAM, J. Alg. Disc. Meth., 2, 311-323 (1981) · Zbl 0499.62037
[21] Major, P., On a non-parametric estimation of the regression function, Stud. Sci. Math. Hungar., 8, 347-361 (1973) · Zbl 0315.62020
[22] Nadaraya, E. A., On estimating regression, Theor. Probab. Appl., 9, 141-142 (1964) · Zbl 0136.40902
[23] Nadaraya, E. A., Remarks on nonparametric estimates for density functions and regression curves, Theor. Probab. Appl., 15, 134-137 (1970) · Zbl 0228.62031
[24] Nadaraya, E. A., Some limit theorems related to nonparametric estimates of regression curve (in Russian), Bull. Acad. Sci. Georgian S.S.R., 71, 57-60 (1973)
[25] Nadaraya, E. A., The limit distribution of the quadratic deviation of nonparametric estimates of the regression function, Soobshch. Akad. Nauk. Gruz. SSR, 74, No. 1, 33-36 (1974)
[26] Noda, K., Estimation of a regression function by the Parzen kernel-type density estimators, Ann. Inst. Statist. Math., 28, No. 2, 221-234 (1976) · Zbl 0369.62068
[27] Parzen, E., On estimation of a probability density function and mode, Ann. Math. Statist., 33, 1065-1076 (1962) · Zbl 0116.11302
[28] Reiss, R.-D., Consistency of a certain class of empirical density functions, Metrika, 22, 189-203 (1975) · Zbl 0322.60021
[29] Révész, P., On strong approximation of the multidimensional empirical process, Ann. Probab., 4, 729-743 (1976) · Zbl 0344.60022
[30] Révész, P., On the nonparametric estimation of the regression function, Problems Control. Inform. Theory, 8, 297-302 (1979) · Zbl 0417.62050
[31] Rosenblatt, M., Remarks on a multivariate transformation, Ann. Math. Statist., 23, 470-472 (1952) · Zbl 0047.13104
[32] Rosenblatt, M., Remarks on some nonparametric estimators of a density function, Ann. Math. Statist., 27, 832-837 (1956) · Zbl 0073.14602
[33] Rosenblatt, M.: Conditional probability density and regression estimates. In Multivariate Analysis II, ed. Krishnaiah, 25-31 (1969)
[34] Rosenblatt, M., On the maximal deviation of a k-dimensional density estimator, Ann. Probab., 4, 1009-1015 (1976) · Zbl 0369.62028
[35] Royall, R.M.: A class of nonparametric estimators of a nonlinear regression function. Ph.D. dissertation, Stanford University (1966)
[36] Schuster, E. F., Joint asymptotic distribution of the estimated regression function at a finite number of distinct points, Ann. Math. Statist., 43, 84-88 (1972) · Zbl 0248.62027
[37] Schuster, E. F.; Yakowitz, S., Contributions to the theory of nonparametric regression, with application to system identification, Ann. Statist., 7, 139-149 (1979) · Zbl 0401.62033
[38] Silverman, B. W., On a Gaussian process related to multivariate probability density estimation, Math. Proc. Cambridge Philos. Soc., 80, 135-144 (1976) · Zbl 0385.60042
[39] Silverman, B. W., Weak and strong uniform consistency of the kernel estimate of a density and its derivatives, Ann. Statist., 6, 177-184 (1978) · Zbl 0376.62024
[40] Stone, C. J., Consistent nonparametric regression, Ann. Statist., 5, 595-620 (1977) · Zbl 0366.62051
[41] Stute, W., A law of logarithm for kernel density estimators, Ann. Probability, 10, 414-422 (1982) · Zbl 0493.62040
[42] Tusnády, G., A remark on the approximation of the sample d.f. in the multidimensional case, Period. Math. Hungar., 8, 53-55 (1977) · Zbl 0386.60006
[43] Wandl, H., On kernel estimation of regression functions, Wiss. Sit. z. Stoch., WSS-03, 1-25 (1980)
[44] Watson, G. S., Smooth regression analysis, Sankhyā Ser. A, 26, 359-372 (1964) · Zbl 0137.13002
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.