Author:
Bloznelis Mindaugas,Götze Friedrich
Abstract
AbstractWe study the distribution of a general class of asymptotically linear statistics which are symmetric functions of N independent observations. The distribution functions of these statistics are approximated by an Edgeworth expansion with a remainder of order $$o(N^{-1})$$
o
(
N
-
1
)
. The Edgeworth expansion is based on Hoeffding’s decomposition which provides a stochastic expansion into a linear part, a quadratic part as well as smaller higher order parts. The validity of this Edgeworth expansion is proved under Cramér’s condition on the linear part, moment assumptions for all parts of the statistic and an optimal dimensionality requirement for the non linear part.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Statistics, Probability and Uncertainty,Statistics and Probability,Analysis
Reference37 articles.
1. Angst, J., Poly, G.: A weak Cramér condition and application to Edgeworth expansions. Electron. J. Probab. 22(59), 1–24 (2017)
2. Babu, G.J., Bai, Z.D.: Edgeworth expansions of a function of sample means under minimal moment conditions and partial Cramer’s condition. Sankhya Ser. A 55, 244–258 (1993)
3. Bai, Z.D., Rao, C.R.: Edgeworth expansion of a function of sample means. Ann. Stat. 19, 1295–1315 (1991)
4. Bentkus, V., Götze, F., van Zwet, W.R.: An Edgeworth expansion for symmetric statistics. Ann. Stat. 25, 851–896 (1997)
5. Bentkus, V., Götze, F.: Lattice point problems and distribution of values of quadratic forms. Ann. Math. (2) 150(3), 977–1027 (1999)