Discrepancy functions considered in the simulations of this paper.

NOTE: denotes a vector of length n, composed of random numbers from the uniform distribution that are independent from each other and from all the other random variables considered. denotes the cumulative distribution function of the standard normal distribution, and denotes the cumulative distribution function of the density of the model – or a randomized version of it when X is discrete:

where is a small positive number so that remains bigger than the closest smaller discrete value to . Normalized data are defined as .
We considered the following t functions: mean, variance, and only in the case of unnormalized data, and maximum (only for comparing with under the Poisson model), and only in the case of normalized data, skewness, kurtosis, and

where denotes the ascending ordered version of . is obtained as , with the likelihood ratio statistic as and an adequate weight function [37]. Centered mean and variance are the empirical mean and variance minus the mean and variance expected with .