General cumulative Kullback–Leibler information

The cumulative residual Kullback–Leibler information is defined on the semi-infinite (non negative) interval. In this paper, we extend the cumulative residual Kullback–Leibler information to the whole real line and propose a general cumulative Kullback–Leibler information. We study its application to a test for normality in comparison with some competing test statistics based on the empirical distribution function including the well-known tests applied in practice like Kolmogorov–Smirnov, Cramer–von Mises, Anderson–Darling, and other existing tests.