The Annals of Statistics

The Empirical Characteristic Function and Its Applications

Andrey Feuerverger and Roman A. Mureika

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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.

Article information

Ann. Statist. Volume 5, Number 1 (1977), 88-97.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G99: None of the above, but in this section
Secondary: 60G99: None of the above, but in this section 62M99: None of the above, but in this section

Empirical characteristic function characteristic function uniform almost sure convergence weak convergence Gaussian processes testing for symmetry asymptotic distribution


Feuerverger, Andrey; Mureika, Roman A. The Empirical Characteristic Function and Its Applications. Ann. Statist. 5 (1977), no. 1, 88--97. doi:10.1214/aos/1176343742.

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