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Some results on increments of the partially observed empirical process
Author(s):
Zacharie
Dindar
Journal:
Trans. Amer. Math. Soc.
353
(2001),
427-440.
MSC (2000):
Primary 60F17;
Secondary 62G07
Posted:
October 23, 2000
MathSciNet review:
1804505
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Abstract:
The author investigates the almost sure behaviour of the increments of the partially observed, uniform empirical process. Some functional laws of the iterated logarithm are obtained for this process. As an application, new laws of the iterated logarithm are established for kernel density estimators.
References:
-
- 1.
- Bosq D. and Lecoutre J.P. (1987). Théorie de l'estimation fonctionnelle. Economie et Statistiques avancées. Economica, Paris.
- 2.
- Chernoff H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics, 23, 493-507. MR 15:241e
- 3.
- Conway J.B. (1985). A course in functional analysis. Springer, Berlin. MR 86h:46001
- 4.
- Deheuvels P. (1992). Functional laws of the iterated logarithm for large increments of empirical and quantile processes. Stochastic Processes and their Applications, 43, 133-163. MR 93m:60057
- 5.
- Deheuvels P. and Mason D.M. (1990). Nonstandard functional laws of the iterated logarithm for tail empirical and quantile processes. Annals of Probability, 18, 1693-1722. MR 91j:60056
- 6.
- Deheuvels P. and Mason D.M. (1992). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Annals of Probability, 20, 1248-1287. MR 93h:60046
- 7.
- Deheuvels P. and Mason D.M. (1994). Random fractals generated by oscillations of processes with stationary and independent increments. Probability in Banach spaces, Sandjberg, 9, 73-89. MR 96b:60191
- 8.
- Deheuvels P. and Mason D.M. (1995). On the fractal nature of empirical increments. Annals of Probability, 23, 355-387. MR 96e:60064
- 9.
- Deheuvels P. and Révész P. (1986). Weak laws for the increments of Wiener processes, Brownian bridges, empirical processes and partial sums of I.I.D.R.V.'s. Mathematical Statistics and Probability theory, Vol. A, Bad Tatzmannsdorf, 69-88. MR 89a:60057
- 10.
- Deuschel J.D. and Stroock D.W. (1989). Large Deviations. Academic Press, New York. MR 90h:62006
- 11.
- Dindar Z. (1998). Some more results on increments of the partially observed empirical process. Submitted
- 12.
- Härdle W. (1991). Smoothing techniques. With implementation in S. Springer, New York. MR 93b:62006
- 13.
- James B.R. (1975). A functional law of the iterated logarithm for weighted empirical distributions. Annals of Probability, 3, 762-772. MR 53:6695
- 14.
- Komlós J., Major P. and Tusnády G. (1975a). Weak convergence and embedding. Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam, 11, 149-165. MR 53:6676
- 15.
- Komlós J., Major P. and Tusnády G. (1975b). An approximation of partial sums of independent r.v.'s and the sample d.f. I. Z. Wahrsch. Verw. Gebiete, 32, 111-131. MR 51:11605b
- 16.
- Komlós J., Major P. and Tusnády G. (1976). An approximation of partial sums of independant r.v.'s and the sample d.f. II. Z. Wahrsch. Verw. Gebiete, 34, 33-58. MR 53:6697
- 17.
- Lifshits M.A. and Weber M. (1997). Strassen laws of iterated logarithm for partially observed processes. Journal of Theoretical Probability, 10, 101-115. MR 98h:60041
- 18.
- Parzen E. (1962). On estimation of a probability density function and mode. Annals of Mathematical Statistics, 33, 1065-1076. MR 26:841
- 19.
- Rosenblatt M. (1979). Global measures of deviation for kernel and nearest neighbor density estimates. Smoothing techniques for curve estimation, Lecture Notes in Mathematics, 757. Springer, Berlin. MR 81g:62076
- 20.
- Schilder M. (1966). Some asymptotic formulae for Wiener integrals. Transactions of the American Mathematical Society, 125, 63-85. MR 34:1770
- 21.
- Shorack G.R. and Wellner J.A. (1986). Empirical processes with applications to statistics. Wiley, New York. MR 88e:60002
- 22.
- Strassen V. (1964). An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete, 3, 211-226. MR 30:5379
- 23.
- Stute W. (1982a). The oscillation behaviour of empirical process. Annals of Probability, 10, 86-107. MR 83e:60028
- 24.
- Stute W. (1982b). A law of the iterated logarithm for kernel density estimators. Annals of Probability, 10, 414-422. MR 84e:62060
- 25.
- Varadhan S.R.S. (1966). Asymptotic probabilities and differential equations. Communications on Pure and Applied Mathematics, 19, 261-286. MR 34:3083
- 26.
- Weber M. (1988). The law of the iterated logarithm for subsequences in Banach spaces. Probability in Banach spaces, 7, 269-285. MR 92h:60009
- 27.
- Weber M. (1990). The law of the iterated logarithm on subsequences-characterizations. Nagoya Mathematical Journal, 118, 65-97. MR 92a:60089
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Additional Information:
Zacharie
Dindar
Affiliation:
L.S.T.A., Université Paris VI, 45 rue Vineuse, 75016 Paris, France
Email:
dindar@ccr.jussieu.fr
DOI:
10.1090/S0002-9947-00-02736-7
PII:
S 0002-9947(00)02736-7
Keywords:
Empirical processes,
law of the iterated logarithm,
functional laws
Received by editor(s):
March 26, 1999
Received by editor(s) in revised form:
April 26, 2000
Posted:
October 23, 2000
Copyright of article:
Copyright
2000,
American Mathematical Society
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