Empirical characteristic function

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

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

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

It is useful in estimation theory and hypothesis testing.