An omnibus test for the two-sample problem using the empirical characteristic function

The empirical characteristic function (CF) is the Fourier transform of the sample distribution function. The values of its real and imaginary parts at some real number t are merely sample means of cosine and sine functions of the data, the observations being multiplied by t. Given independent samples from two populations, we develop a test for the two-sample problem which is based on a quadratic form in differences between the respective components of the empirical CFs of the two samples. The power of the CF test compares favorably with that of competing omnibus tests when the data are continuous. In the discrete case the CF procedure is also applicable and quite successful; and in this application it appears to have no competitors.