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We propose a new powerful family of tests of univariate normality. Thesetests are based on an initial value problem in the space of characteristicfunctions originating from the fixed point property of the normal distributionin the zero bias transform. Limit distributions of the test statistics areprovided under the null hypothesis, as well as under contiguous and fixedalternatives. Using the covariance structure of the limiting Gaussian processfrom the null distribution, we derive explicit formulas for the first fourcumulants of the limiting random element and apply the results by fitting adistribution from the Pearson system. A comparative Monte Carlo power studyshows that the new tests are serious competitors to the strongest wellestablished tests.

On combining the zero bias transform and the empirical characteristic function to test normality