The Empirical Characteristic Function and Its Applications

Abstract

Certain probability properties of cn(t), the empirical characteristic function $(\operatorname{ecf})$ are investigated. More specifically it is shown under some general restrictions that cn(t) converges uniformly almost surely to the population characteristic function c(t). The weak convergence of n1/2(cn(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 ∫[Im cn(t)]2 dG(t) is proposed and its asymptotic distribution evaluated.