Distributions of the Kullback–Leibler divergence with applications

The Kullback-Leibler divergence (KLD) is a widely used method for measuring the fit of two distributions. In general, the distribution of the KLD is unknown. Under reasonable assumptions, common in psychometrics, the distribution of the KLD is shown to be asymptotically distributed as a scaled (non-central) chi-square with one degree of freedom or a scaled (doubly non-central) F. Applications of the KLD for detecting heterogeneous response data are discussed with particular emphasis on test security.