Testing Gaussianity with the characteristic function

We wish to formulate a test for the hypothesis X_i~N(μ,σ²) for i=0,1,...,N-1 against unspecified alternatives. We assume independence of the components of X=[X₀,X₁,...,X_N-1]. This is a problem of universal importance as the assumption of Gaussianity is prevalent and fundamental to many statistical theories and engineering applications. Many such tests exist, the most well-known being the χ ² goodness-of-fit test with its variants and the Kolmogorov-Smirnov one-sample cumulative probability function test. More powerful modern tests for the hypothesis of Gaussianity include the D´Agostino (1977) K² and Shapiro-Wilk (1968) W tests. Tests for Gaussianity have been proposed which use the characteristic function. It is the purpose of this paper to highlight and resolve problems with these tests and to improve performance so that the test is competitive with, and in some cases better than, the most powerful known tests for Gaussianity