Abstract
In a previous paper we introduced a notion of a local empirical process indexed by functions which has useful applications to density and regression function estimation among other areas. We have shown that if the function class is uniformly bounded, one can obtain a strong approximation to this process by a suitable Gaussian process. We now prove such a result when the underlying function class is unbounded, but has an envelope function with a finite p-th moment for some p > 2. Among other applications, our new strong invariance principle for the unbounded case can be used to prove laws of the iterated logarithm for the kernel regression function estimator under mild conditions.
Research partially supported by the Volkswagenstiftung (RiP program at Oberwolfach) and an NSF Grant.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Deheuvels, P. and Mason, D.M. (1994). Functional laws of the iterated logarithm for local empirical processes indexed by sets. Ann. Probab. 22 1619–1661.
Deheuvels, P. and Mason, D.M. (1995). Nonstandard local empirical processes indexed by sets. J. Stat. Plan. Inf. 45 91–112.
Dudley, R.M. (1984). A course on empirical processes (École d’Été de Probabilités de Saint-Flour XII-1982) Lecture Notes in Mathematics 1097 2–141, (ed. P.L. Hennequin) Springer-Verlag, New York
Einmahl, U. (1987). A useful estimate in the multidimensional invariance principle.Probab. Th. Re. Fields 76 81–101.
Einmahl, U. (1993). Toward a general law of the iterated logarithm in Banach space. Ann. Probab. 21 2012–2045.
Einmahl, U. and MASON, D.M. (1997). Gaussian approximation of local empirical processes indexed by functions. Probab. Th. Re. Fields 107 283–301.
Giné, E. and ZINN, J. (1984). Some limit theorems for empirical processes. Ann. Probab. 12 929–989.
Haerdle, W. (1984). A law of the iterated logarithm for nonparametric regression function estimators. Ann. Statist. 12 624–635.
Hall, P. (1981). Laws of the iterated logarithm for nonparametric density estimators. Z. Wahrsch. verw. Gebiete. 56 47–61.
Hong, S. Y. (1992). A law of the iterated logarithm for neighour-type regression function estimator. Chin. Ann. Math. 13 180–195.
Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent rv’s and the sample df I. Z. Wahrsch. verw. Gebiete. 32 111–131.
Ledoux, M. and Talagrand, M. (1991). Probability on Banach Spaces, Springer-Verlag, Berlin.
Mack, Y.P. and Silverman, B.W. (1982). Weak and strong uniform consistency of kernel regression estimates. Z. Wahrsch. verw. Gebiete. 61 405–415.
Major, P. (1976). Approximations of partial sums of i.i.d. r.v.s when the summands have only two moments. Z. Wahrscheinlichkeitstheorie verw. Gebiete 35 221–229.
Nadaraya, E.A. (1964). On estimating regression. Theor. Probab. Appl. 9 141–142.
Pollard, D. (1982). Rates of strong uniform convergence, Preprint
Pollard, D. (1984). Convergence of Stochastic Processes. Springer-Verlag, New York.
Tusnády, G. (1977). A remark on the approximation of the sample df in the multidimensional case. Period. Math. Hung. 8 53–55.
Van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes. Springer-Verlag, New York.
Watson, G.S. (1964). Smooth regression analysis. Sankhya A 26 359–372.
Wichura, M.J. (1974). Some Strassen-type laws of the iterated logarithm for multiparameter stochastic processes with independent increments. Ann. Probab. 1 272–296.
Zaitsev, A.Yu. (1987). On the Gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein’s inequality conditions. Probab. Th. Re. Fields 74, 534–566.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Basel AG
About this paper
Cite this paper
Einmahl, U., Mason, D.M. (1998). Strong Approximations to the Local Empirical Process. In: Eberlein, E., Hahn, M., Talagrand, M. (eds) High Dimensional Probability. Progress in Probability, vol 43. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8829-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8829-5_5
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9790-7
Online ISBN: 978-3-0348-8829-5
eBook Packages: Springer Book Archive