The distinct flavors of Zipf's law in the rank-size and in the size-distribution representations, and its maximum-likelihood fitting

In the last years, researchers have realized the difficulties of fitting power-law distributions properly. These difficulties are higher in Zipf's systems, due to the discreteness of the variables and to the existence of two representations for these systems, i.e., two versions about which is the random variable to fit. The discreteness implies that a power law in one of the representations is not a power law in the other, and vice versa. We generate synthetic power laws in both representations and apply a state-of-the-art fitting method (based on maximum-likelihood plus a goodness-of-fit test) for each of the two random variables. It is important to stress that the method does not fit the whole distribution, but the tail, understood as the part of a distribution above a cut-off that separates non-power-law behavior from power-law behavior. We find that, no matter which random variable is power-law distributed, the rank-size representation is not adequate for fitting, whereas the representation in terms of the distribution of sizes leads to the recovery of the simulated exponents, may be with some bias.