## The Annals of Statistics

### The Empirical Characteristic Function and Its Applications

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176343742

Digital Object Identifier
doi:10.1214/aos/1176343742

Mathematical Reviews number (MathSciNet)
MR428584

Zentralblatt MATH identifier
0364.62051

JSTOR