Empirical characteristic function

In statistics, the empirical characteristic function is a numerically calculated approximation to the characteristic function.

If $X_{i},i=1,2,\ldots ,n$ are i.i.d. observations, then the empirical characteristic function $\varphi _{n}(t)$ is defined as

$\varphi _{n}(t)={\frac {1}{n}}\sum _{j=1}^{n}\exp(itX_{j}).$

It is useful in estimation theory and hypothesis testing.