A new class of binning free, multivariate goodness-of-fit tests: the energy tests

We present a new class of multivariate binning-free and nonparametric goodness-of-fit tests. The test quantity \emph{energy} is a function of the distances of observed and simulated observations in the variate space. The simulation follows the probability distribution function $f_{0}$ of the null hypothesis. The distances are weighted with a weighting function which can be adjusted to the variations of $f_{0}$. We have investigated the power of the test for a uniform and a Gaussian distribution of one or two variates, respectively and compared it to that of conventional tests. The energy test with a Gaussian weighting function is closely related to the Pearson $\chi ^{2}$ test but is more powerful in most applications and avoids arbitrary bin boundaries. The test is especially powerful in the multivariate case.