## The Annals of Statistics

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

### The Empirical Characteristic Function and Its Applications

Andrey Feuerverger and Roman A. Mureika

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

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

**Keywords**

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

#### Citation

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