Related papers: Parameter estimation for power-law distributions b…
Many experimental quantities show a power-law distribution $p(x)\propto x^{-\alpha}$. In astrophysics, examples are: size distribution of dust grains or luminosity function of galaxies. Such distributions are characterized by the exponent…
Maximum likelihood estimations for the parameters of extreme value distributions are discussed in this paper using fixed point iteration. The commonly used numerical approach for addressing this problem is the Newton-Raphson approach which…
Over the last few decades power law distributions have been suggested as forming generative mechanisms in a variety of disparate fields, such as, astrophysics, criminology and database curation. However, fitting these heavy tailed…
This expository note describes how to apply the method of maximum likelihood to estimate the parameters of the ``$q$-exponential'' distributions introduced by Tsallis and collaborators. It also describes the relationship of these…
We review briefly the concepts underlying complex systems and probability distributions. The later are often taken as the first quantitative characteristics of complex systems, allowing one to detect the possible occurrence of regularities…
The maximum likelihood principle is widely used in statistics, and the associated estimators often display good properties. indeed maximum likelihood estimators are guaranteed to be asymptotically efficient under mild conditions. However in…
In this article, we proposed a new probability distribution named as power Maxwell distribution (PMaD). It is another extension of Maxwell distribution (MaD) which would lead more flexibility to analyze the data with non-monotone failure…
The so-called Pareto-Levy or power-law distribution has been successfully used as a model to describe probabilities associated to extreme variations of worldwide stock markets indexes data and it has the form $Pr(X>x) ~ x**(-alpha) for…
A simple fragmentation model is introduced and analysed. We show that, under very general conditions, an effective power law for the mass distribution arises with realistic exponent. This exponent has a universal limit, but in practice the…
Many man-made and natural phenomena, including the intensity of earthquakes, population of cities and size of international wars, are believed to follow power-law distributions. The accurate identification of power-law patterns has…
The main approach to inference for multivariate extremes consists in approximating the joint upper tail of the observations by a parametric family arising in the limit for extreme events. The latter may be expressed in terms of…
This short communication uses a simple experiment to show that fitting to a power law distribution by using graphical methods based on linear fit on the log-log scale is biased and inaccurate. It shows that using maximum likelihood…
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…
This paper generalizes the notion of sufficiency for estimation problems beyond maximum likelihood. In particular, we consider estimation problems based on Jones et al. and Basu et al. likelihood functions that are popular among…
Empirical distributions have their in-sample maxima as natural censoring. We look at the "hidden tail", that is, the part of the distribution in excess of the maximum for a sample size of $n$. Using extreme value theory, we examine the…
In this paper, insight is given in the techniques used to compute asymptotic expansions. In a broad fashion the technique is described. Most of the results apply to the paper "An expansion for the maximum likelihood estimator and its…
For a large class of statistical systems a geometric mean value of the observables is constrained. These observables are characterized by a power-law statistical distribution.
Probability distributions having power-law tails are observed in a broad range of social, economic, and biological systems. We describe here a potentially useful common framework. We derive distribution functions $\{p_k\}$ for situations in…
Loss tomography has been studied for more than 10 years and a number of estimators have been proposed. The estimators can be divided into two classes: maximum likelihood and non-maximum likelihood. The maximum likelihood estimators rely on…
Selected examples of possible origins of power-law distributions are presented.