A test for normality and independence based on characteristic function

In this article we prove a generalization of the Ejsmont characterization (Ejsmont in Stat Probab Lett 114:1–5, 2016) of the multivariate normal distribution. Based on it, we propose a new test for independence and normality. The test uses an integral of the squared modulus of the difference between the product of empirical characteristic functions and some constant. Special attention is given to the case of testing for univariate normality in which we derive the test statistic explicitly in terms of Bessel function and explore asymptotic properties. The simulation study also includes the cases of testing for bivariate and trivariate normality and independence, as well as multivariate normality. We show the quality performance of our test in comparison to some popular powerful competitors. The practical application of the proposed normality and independence test is discussed and illustrated using a real dataset.