Certain probability properties of $c_n(t)$, the empirical characteristic function $(\operatorname{ecf})$ are investigated. More specifically it is shown under some general restrictions that $c_n(t)$ converges uniformly almost surely to the population characteristic function $c(t).$ The weak convergence of $n^{\frac{1}{2}}(c_n(t) - c(t))$ to a Gaussian complex process is proved. It is suggested that the ecf may be a useful tool in numerous statistical problems. Application of these ideas is illustrated with reference to testing for symmetry about the origin: the statistic $\int\lbrack\mathbf{Im} c_n(t)\rbrack^2 dG(t)$ is proposed and its asymptotic distribution evaluated.
Publié le : 1977-01-14
Classification:
Empirical characteristic function,
characteristic function,
uniform almost sure convergence,
weak convergence,
Gaussian processes,
testing for symmetry,
asymptotic distribution,
62G99,
60G99,
62M99
@article{1176343742,
author = {Feuerverger, Andrey and Mureika, Roman A.},
title = {The Empirical Characteristic Function and Its Applications},
journal = {Ann. Statist.},
volume = {5},
number = {1},
year = {1977},
pages = { 88-97},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343742}
}
Feuerverger, Andrey; Mureika, Roman A. The Empirical Characteristic Function and Its Applications. Ann. Statist., Tome 5 (1977) no. 1, pp. 88-97. http://gdmltest.u-ga.fr/item/1176343742/