exponential cut-off; flexible modelling; Pareto distribution; power law; Weibull distribution; Statistics and Probability; Statistics, Probability and Uncertainty
Abstract :
[en] Power laws and power laws with exponential cut-off are two distinct families of distributions on the positive real half-line. In the present paper, we propose a unified treatment of both families by building a family of distributions that interpolates between them, which we call Interpolating Family (IF) of distributions. Our original construction, which relies on techniques from statistical physics, provides a connection for hitherto unrelated distributions like the Pareto and Weibull distributions, and sheds new light on them. The IF also contains several distributions that are neither of power law nor of power law with exponential cut-off type. We calculate quantile-based properties, moments and modes for the IF. This allows us to review known properties of famous distributions on (Formula presented.) and to provide in a single sweep these characteristics for various less known (and new) special cases of our Interpolating Family.
Disciplines :
Mathematics
Author, co-author :
Sinner, Corinne; Département de Mathématiques, Université libre de Bruxelles CP210, Boulevard du Triomphe, Bruxelles, Belgium
Dominicy, Yves; Bank employee, Luxembourg
Trufin, Julien ; Département de Mathématiques, Université libre de Bruxelles CP210, Boulevard du Triomphe, Bruxelles, Belgium
Waterschoot, Wout; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium
Weber, Patrick; Département de Mathématiques, Université libre de Bruxelles CP210, Boulevard du Triomphe, Bruxelles, Belgium
LEY, Christophe ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium
External co-authors :
yes
Language :
English
Title :
From Pareto to Weibull – A Constructive Review of Distributions on ℝ+
Aban, I.B., Meerschaert, M.M. & Panorska, A.K. (2006). Parameter estimation for the truncated Pareto distribution. J. Am. Stat. Assoc., 101, 270–277.
Adamic, L.A. & Huberman, B.A. (2000). The nature of markets in the world wide web. Q. J. Electron. Commer., 1, 512.
Aiello, W., Chung, F. & Lu, L. (2000). A random graph model for massive graphs. In Proceedings of the thirty-second annual acm symposium on theory of computing, STOC '00, Association for Computing Machinery: New York, NY, USA, pp. 171–180.
Aschwanden, M. (2013). SOC systems in astrophysics. In Self-organized critical phenomena, Ed. Aschwanden, M., Open Academic Press: Berlin.
Auberbach, F. (1913). Das Gesetz der Bevölkerungskonzentration. Petermanns Geogr., 59, 74–76.
Babić, S., Ley, C. & Veredas, D. (2019). Comparison and classification of flexible distributions for multivariate skew and heavy-tailed data. Symmetry, 11(10), 1216.
Burroughs, S.M. & Tebbens, S.F. (2001). Upper-truncated power laws in natural systems. Pure Appl. Geophys., 158, 741–757.
Clarke, R.T. (2002). Estimating trends in data from the Weibull and a generalized extreme value distribution. Water Resour. Res., 38, 25–1–25–10.
Clauset, A., Shalizi, C.R. & Newman, M.E.J. (2009). Power-law distributions in empirical data. SIAM Rev., 51, 661–703.
Crovella, M.E. & Bestavros, A. (1997). Self-similarity in world wide web traffic: Evidence and possible causes. IEEE/ACM Trans. Networking, 5(6), 835–846.
de Solla Price, D.J. (1965). Networks of scientific papers. Science, 149, 510–515.
Dominicy, Y. & Sinner, C. (2017). Distributions and composite models for size-type data. In Advances in Statistical Methodologies and Their Application to Real Problems, InTechOpen, pp. 159.
Eeckhout, J. (2004). Gibrat's Law for (All) Cities. Am. Econ. Rev., 94, 1429–1451.
Estoup, J.B. (1916). Gammes Stenographiques. Institut Stenographique de France, Paris.
Farmer, J.D. & Geanakoplos, J. (2008). Power laws in economics and elsewhere.
Fréchet, M. (1927). Sur la loi de probabilité de l'écart maximum. Ann. Soc. Pol. Math. Cracovie, 6, 93–116.
Fujisawa, H. & Abe, T. (2015). A family of skew distributions with mode-invariance through transformation of scale. Stat. Methodol., 25, 89–98.
Gabaix, X. (2016). Power Laws in Economics: An Introduction. J. Econ. Perspect., 30, 185–206.
Huberman, B.A. & Adamic, L.A. (1999). Growth Dynamics of the World Wide Web. Nature, 401, 131.
Huberman, B.A. & Adamic, L.A. (2004). Information dynamics in the networked world. In Complex networks, Eds. Ben-Naim, E., Frauenfelder, H. & Toroczkai, Z., Vol. 650, Springer, pp. 371–398.
Jones, M.C. (2014). Generating distributions by transformation of scale. Stat. Sin., 24, 749–772.
Jones, M.C. (2015). On families of distributions with shape parameters (with discussion). Int. Stat. Rev., 83, 175–192.
Kato, S. & Jones, M.C. (2015). A tractable and interpretable four-parameter family of unimodal distributions on the circle. Biometrika, 102, 181–190.
Kleiber, C. & Kotz, S. (2003). Statistical size distributions in economics and actuarial sciences. John Wiley & Sons: New York.
Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, Wiley Series in Probability and Statistics. Wiley.
Lee, E.T. & Wang, J. (2003). Statistical Methods for Survival Data Analysis, Wiley Series in Probability and Statistics. Wiley.
Ley, C. (2015). Flexible modelling in statistics: past, present and future. J. Soc. Française de Stat., 156, 76–96.
Ley, C., Babić, S. & Craens, D. (2021). Flexible models for complex data with applications. Ann. Rev. Stat. Appl., 8, 18.1–18.23.
Ley, C. & Simone, R. (2020). Modelling earthquakes: Characterizing magnitudes and inter-arrival times. In Computational and methodological statistics and biostatistics, Springer, pp. 29–50.
Lu, E.T. & Hamilton, R.J. (1991). Avalanches of the distribution of solar flares. Astrophys. J., 380, 89–92.
Luttmer, E. (2007). Selection, Growth, and the Size Distribution of Firms. Q. J. Econ., 122, 1103–1144.
Mandelbrot, B. (1964). Random Walks, Fire Damage Amount and Other Paretian Risk Phenomena. Oper. Res., 12, 582–585.
Mandelbrot, B.B. (1983). The Fractal Geometry of Nature. New York: W. H. Freeman.
Manwell, J.F., McGowan, J.G. & Rogers, A.L. (2009). Wind energy explained: Theory, design and application. Wiley: Chichester.
Marchenko, Y.V. & Genton, M.G. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data. Environmetrics, 21, 318–340.
Meidell, B. (1912). Zur Theorie des Maximums. Septième Congrès d'Actuaires, 1, 85–99.
Mitzenmacher, M. (2004). A Brief History of Generative Models for Power Law and Lognormal Distributions. Internet Math., 1, 226–251.
Miyazima, S., Lee, Y., Nagamine, T. & Miyajima, H. (2000). Power-law distribution of family names in japanese societies. Physica A, 278, 282–288.
Muñoz, M.A. (2018). Colloquium: Criticality and dynamical scaling in living systems. Rev. Mod. Phys., 90, 31001.
Nelson, W.B. (2005). Applied life data analysis, Wiley Series in Probability and Statistics. Wiley.
Neukum, G. & Ivanov, B.A. (1994). Crater size distributions and impact probabilities on earth from lunar, terrestrial planeta, and asteroid cratering data. Hazards Due to Comets and Asteroids, 359–416.
Newman, M.E.J. (2005). Power laws, Pareto distributions and Zipf's law. Contemp. Phys., 46, 323–351.
Pareto, V. (1897). Cours d'économie politique, tome second. Lausanne: F. Rouge, Libraire-Éditeur.
Piketty, T. & Zucman, G. (2014). Capital is Back: Wealth-Income Ratios in Rich Countries, 1700-2010. Q. J. Econ., 129, 1255–1310.
Rausand, M. & Høyland, A. (2004). System Reliability Theory: Models, Statistical Methods, and Applications, Wiley Series in Probability and Statistics. Wiley.
Reed, W.J.R. & Jorgensen, M. (2004). The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions. Commun. Stat. – Theory Methods, 33, 1733–1753.
Roberts, D.C. & Turcotte, D.L. (1998). Fractality and selforganized criticality of wars. Fractals, 6, 351–357.
Rosen, S. (1981). The Economics of Superstars. Am. Econ. Rev., 71, 845–858.
Rosin, P. & Rammler, E. (1933). The laws governing the fineness of powdered coal. J. Inst. Fuel, 7, 29–36.
Small, M. & Singer, J.D. (1982). Resort to arms: International and civil wars, 1816-1980. Sage Publications: Beverley Hills.
Sornette, D. (2003). Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization, and Disorder: Concepts and Tools, 2. Springer: Heidelberg.
Toda, A.A. & Walsh, K. (2015). The Double Power Law in Consumption and Implications for Testing Euler Equations. J. Polit. Econ., 123, 1177–1200.
Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys., 52, 479–487.
Tsallis, C. (2002). Nonextensive statistical mechanics: a brief review of its present status. An. Acad. Bras Cienc., 74, 393–414.
Turcotte, D.L. (1997). Fractals and Chaos in Geology and Geophysics, 2. Cambridge University Press: Cambridge.
Weibull, W. (1939). A statistical theory of the strength of materials. R. Inst. Engeneering Res., 151, 1–45.
Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech., 18, 290–293.
White, E.P., Enquist, B.J. & Green, J.L. (2008). On estimating the exponent of power law frequency distributions. Ecology, 89, 905–912.
Zipf, G.K. (1949). Human Behaviour and the Principle of Least Effort. Addison-Wesley, Reading, MA.
