q-exponential, Weibull, and q-Weibull distributions: an empirical analysis

Abstract

In a comparative study, the q-exponential and Weibull distributions are employed to investigate frequency distributions of basketball baskets, cyclone victims, brand-name drugs by retail sales, and highway length. In order to analyze the intermediate cases, a distribution, the q-Weibull one, which interpolates the q-exponential and Weibull ones, is introduced. It is verified that the basketball baskets distribution is well described by a q-exponential, whereas the cyclone victims and brand-name drugs by retail sales ones are better adjusted by a Weibull distribution. On the other hand, for highway length the q-exponential and Weibull distributions do not give satisfactory adjustment, being necessary to employ the q-Weibull distribution. Furthermore, the introduction of this interpolating distribution gives an illumination from the point of view of the stretched exponential against inverse power law (q-exponential with q > 1) controversy. 89.90.+n

q-distributions in complex systems: a brief review

“…q-exponential distributions have also been applied in the empirical study of stock markets [39][40][41][42], DNA sequences [43], family names [44], human behavior [45][46][47], geomagnetic records [48,49], train delays [50], reaction kinetics [51], air networks [52], hydrological phenomena [53], fossil register [54], basketball [55], earthquakes [56][57][58], world track records [59], voting processes [60], internet [61], individual success [62], citations of scientific papers [63,64], football [65], linguistics [66,67] and solar neutrinos [68,69].…”

“…For example, new classes of generalized asymmetric distributions have been introduced which include q-Weibull as a special case [117,118]. q-Weibull has also been applied in the study of fractal kinetics [119], dieletric breakdown in oxides [120], relaxation in heterogeneous systems [121], ciclone victims and highway lengths [55] among others. …”

“…For example, this basic procedure applied in standard exponential, Gaussian and Weibull distributions leads to q-exponential, q-Gaussian and q-Weibull, respectively [55]. This viewpoint suggests the consideration of other q-distributions which could be obtained by simply replacing its exponential function by a q-exponential one.…”

On superstatistical multiplicative-noise processes

“…[21] we shall call PDF (24) q-Weibull distribution. The q-Weibull distribution has been used to numerically adjust a variety of PDFs, but, to the best of my knowledge, no dynamical basis has been presented so far.…”

Combining Generalized Renewal Processes with Non-Extensive Entropy-Based q-Distributions for Reliability Applications

“…The q-Weibull is a probability distribution that smoothly interpolates the q-Exponential and the Weibull distributions to generate a unified framework to accommodate different cases of data adjustment [15]. Indeed, the q-Weibull distribution has the Exponential, Weibull and q-Exponential distributions as special cases and the q-Exponential generalizes the Exponential probability model.…”