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January, 1977 The Empirical Characteristic Function and Its Applications
Andrey Feuerverger, Roman A. Mureika
Ann. Statist. 5(1): 88-97 (January, 1977). DOI: 10.1214/aos/1176343742


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.


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Andrey Feuerverger. Roman A. Mureika. "The Empirical Characteristic Function and Its Applications." Ann. Statist. 5 (1) 88 - 97, January, 1977.


Published: January, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0364.62051
MathSciNet: MR428584
Digital Object Identifier: 10.1214/aos/1176343742

Primary: 62G99
Secondary: 60G99 , 62M99

Keywords: asymptotic distribution , Characteristic function , Empirical characteristic function , Gaussian processes , testing for symmetry , uniform almost sure convergence , weak convergence

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 1 • January, 1977
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