Empirical characteristic function

Let ${\displaystyle (X_{1},...,X_{n})}$ be independent, identically distributed real-valued random variables with common characteristic function ${\displaystyle \varphi (t)}$. The empirical characteristic function (ECF) defined as

${\displaystyle \varphi _{n}(t)={\frac {1}{n}}\sum _{j=1}^{n}e^{{\rm {i}}tX_{j}},\ {\rm {i}}={\sqrt {-1}},}$

is an unbiased and consistent estimator of the corresponding population characteristic function ${\displaystyle \varphi (t)}$, for each ${\displaystyle t\in \mathbb {R} }$. The ECF apparently made its debut in page 342 of the classical textbook of Cramér (1946),[1] and then as part of the auxiliary tools for density estimation in Parzen (1962).[2] Nearly a decade later the ECF features as the main object of research in two separate lines of application: In Press (1972)[3] for parameter estimation and in Heathcote (1972)[4] for goodness-of-fit testing. Since that time there has subsequently been a vast expansion of statistical inference methods based on the ECF. For reviews of estimation methods based on the ECF the reader is referred to Csörgő (1984a),[5] Rémillard and Theodorescu (2001),[6] Yu (2004),[7] and Carrasco and Kotchoni (2017),[8] while testing procedures are surveyed by Csörgő (1984b),[9] Hušková and Meintanis (2008a),[10] Hušková and Meintanis (2008b),[11] and Meintanis (2016).[12] Ushakov (1999)[13] is also a good source of information regarding the limit properties of the ECF process, as well as on estimation and goodness-of-fit testing via the ECF. A line of research that deserves special mention is ECF testing for independence by means of distance correlation as originally suggested by Székely et al. (2007).[14] This approach has become extremely popular and is currently under vigorous development. We refer to Edelmann et al. (2019)[15] for a recent survey on distance correlation methods.