×

A generalization of the power law distribution with nonlinear exponent. (English) Zbl 1473.60038

Summary: The power law distribution is usually used to fit data in the upper tail of the distribution. However, commonly it is not valid to model data in all the range. In this paper, we present a new family of distributions, the so-called Generalized Power Law (GPL), which can be useful for modeling data in all the range and possess power law tails. To do that, we model the exponent of the power law using a non-linear function which depends on data and two parameters. Then, we provide some basic properties and some specific models of that new family of distributions. After that, we study a relevant model of the family, with special emphasis on the quantile and hazard functions, and the corresponding estimation and testing methods. Finally, as an empirical evidence, we study how the debt is distributed across municipalities in Spain. We check that power law model is only valid in the upper tail; we show analytically and graphically the competence of the new model with municipal debt data in the whole range; and we compare the new distribution with other well-known distributions including the Lognormal, the Generalized Pareto, the Fisk, the Burr type XII and the Dagum models.

MSC:

60E05 Probability distributions: general theory
62F10 Point estimation

Software:

plfit; gbs; optimx; LBFGS-B; R
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Pinto, C. M.; Lopes, A. M.; Machado, J. T., A review of power laws in real life phenomena, Commun Nonlinear Sci Numer Simulat, 17, 9, 3558-3578 (2012) · Zbl 1248.60020
[2] Clauset, A.; Shalizi, C. R.; Newman, M. E., Power-law distributions in empirical data, SIAM Rev, 51, 4, 661-703 (2009) · Zbl 1176.62001
[3] Newman, M. E., Power laws, Pareto distributions and Zipf’s law, Contemp Phys, 46, 5, 323-351 (2005)
[4] Clementi, F.; Di Matteo, T.; Gallegati, M., The power-law tail exponent of income distributions, Phys A: Stat Mech Its Appl, 370, 1, 49-53 (2006)
[5] Rosas-Casals, M.; Solé, R., Analysis of major failures in Europe’s power grid, Int J Electr Power Energy Syst, 33, 3, 805-808 (2011)
[6] Mayo, D. G.; Cox, D. R., Frequentist statistics as a theory of inductive inference, Lecture Notes-Monograph Series, 77-97 (2006) · Zbl 1268.62006
[7] Kelton, W. D.; Averill, M. L., Simulation modeling and analysis (2000), McGraw Hill: McGraw Hill Boston
[8] Prieto, F.; Sarabia, J. M.; Sáez, A. J., Modelling major failures in power grids in the whole range, Int J Electr Power Energy Syst, 54, 10-16 (2014)
[9] Cuadra, L.; Salcedo-Sanz, S.; Del Ser, J.; Jiménez-Fernández, S.; Geem, Z. W., A critical review of robustness in power grids using complex networks concepts, Energies, 8, 9, 9211-9265 (2015)
[10] Stumpf, M. P.; Ingram, P. J.; Nouvel, I.; Wiuf, C., Statistical model selection methods applied to biological networks, Transactions on Computational Systems Biology III, 65-77 (2005), Springer: Springer Berlin Heidelberg · Zbl 1151.62361
[11] Arnold, B. C., Pareto distributions (1983), International Co-operative Publishing House: International Co-operative Publishing House Fairland, Maryland · Zbl 1169.62307
[12] Arnold, B. C., Pareto distributions, second edition, Monographs on statistics and applied probability, 140 (2015), CRC Press. Taylor & Francis Group · Zbl 1361.62004
[13] Sarabia, J. M.; Prieto, F., The pareto-positive stable distribution: a new descriptive model for city size data, Phys A: Stat Mech Appl, 388, 19, 4179-4191 (2009)
[14] Pareto, V., Cours d’économie politique, Librairie Droz (1964)
[15] Guillén, M.; Prieto, F.; Sarabia, J. M., Modelling losses and locating the tail with the pareto positive stable distribution, Insurance, 3, 49, 454-461 (2011)
[16] Arnold, B. C., Pareto and generalized pareto distributions, Modeling income distributions and Lorenz curves, 119-145 (2008), Springer: Springer New York · Zbl 1151.91638
[17] Arnold, B. C., Univariate and multivariate Pareto models, J Stat Distr Appl, 1, 1, 1-16 (2014) · Zbl 1329.62061
[18] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling extremal events, vol. 33 (1997), Springer Science & Business Media · Zbl 0873.62116
[19] Focardi, S. M.; Fabozzi, F. J., The mathematics of financial modeling and investment management, vol. 138 (2004), John Wiley & Sons
[20] Sornette, D., Critical phenomena in natural sciences: chaos, fractals, selforganization and disorder: concepts and tools, Springer Series in Synergetics (2006) · Zbl 1094.82001
[21] Resnick, S. I., Heavy-tail phenomena: probabilistic and statistical modeling (2007), Springer Science & Business Media · Zbl 1152.62029
[22] Castillo, E., Extreme value theory in engineering (2012), Elsevier
[23] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss models: from data to decisions, vol. 715 (2012), John Wiley & Sons · Zbl 1272.62002
[24] Le Courtois, O.; Walter, C., Extreme financial risks and asset allocation (2014), World Scientific Books · Zbl 1298.91010
[25] Castillo, E.; Galambos, J.; Sarabia, J. M., The selection of the domain of attraction of an extreme value distribution from a set of data, (Hüsler, J.; Reiss, R. D., Extreme value theory. Lecture notes in statistics, 51 (1989), Springer), 181-190 · Zbl 0672.62035
[26] Bassi, F.; Embrechts, P.; Kafetzaki, M., Risk management and quantile estimation, (Adler, R.; Feldman, F.; Taqqu, M., A practical guide to heavy tails (1998), Birkhäuser: Birkhäuser Boston), 111-130 · Zbl 0922.62107
[27] Asimit, A. V.; Jones, B. L., Asymptotic tail probabilities for large claims reinsurance of a portfolio of dependent risks, Astin Bull., 38, 147-159 (2008) · Zbl 1169.91361
[28] Cirillo, P., Are your data really pareto distributed?, Phys A: Stat Mech Appl, 392, 23, 5947-5962 (2013) · Zbl 1395.62006
[29] Nair, J.; Wierman, A.; Zwart, B., The fundamentals of heavy-tails: properties, emergence, and identification, Proceedings of the Acm sigmetrics international conference on measurement and modeling of computer systems, 41, 387-388 (2013)
[31] Resnick, S. I., Extreme values, regular variation and point processes (2013), Springer
[32] Von Mises, R., La distribution de la plus grande de n valeurs, Rev Math Union Interbalcanique, 1, 1 (1936) · Zbl 0016.12801
[33] Tyszer, J., Object-oriented computer simulation of discrete-event systems, Vol. 10 (2012), Springer Science & Business Media
[34] Glasserman, P., Monte Carlo methods in financial engineering (Vol. 53) (2003), Springer Science & Business Media
[35] Fisher, R. A., On the mathematical foundations of theoretical statistics, Philos Trans Roy Soc Ser A, 222, 309-368 (1922) · JFM 48.1280.02
[36] Team, R. D.C., R: A language and environment for statistical computing, R foundation for statistical computing, Vienna, Austria (2011)
[37] Nash, J. C.; Varadhan, R., Unifying optimization algorithms to aid software system users: optimx for R, J Stat Softw, 43, 9, 1-14 (2011)
[38] Byrd, R. H.; Lu, P.; Nocedal, J.; Zhu, C. Y., A limited memory algorithm for bound constrained optimization, SIAM J Sci Comput, 16, 5, 1190-1208 (1995) · Zbl 0836.65080
[39] Barros, M.; Paula, G. A.; Leiva, V., An R implementation for generalized Birnbaum-Saunders distributions, Comput Stat Data Anal, 53, 1511-1528 (2009) · Zbl 1452.62012
[40] Lange, K., Numerical analysis for statisticians (2000), Springer: Springer New York
[41] Akaike, H., A new look at the statistical model identification, IEEE Trans Autom Contr, 19, 716-723 (1974) · Zbl 0314.62039
[42] Schwarz, G., Estimating the dimension of a model, Ann Stat, 5, 461-464 (1978) · Zbl 0379.62005
[43] Efron, B., Bootstrap methods: another look at the jackknife, Ann Stat, 7, 1, 1-26 (1979) · Zbl 0406.62024
[44] Wang, C.; Zeng, B.; Shao, J., Application of bootstrap method in Kolmogorov-Smirnov test, Quality, reliability, risk, maintenance, and safety engineering (ICQR2MSE) (2011)
[45] Babu, G. J.; Rao, C. R., Goodness-of-fit tests when parameters are estimated, Sankhya, 66, 63-74 (2004) · Zbl 1192.62126
[46] Kolmogorov, A. N., Sulla determinazione empirica di una legge di distribuzione, Giornale dell’Istituto degli Attuari, 4, 83-91 (1933) · JFM 59.1166.03
[47] Smirnov, N., On the estimation of the discrepancy between empirical curves of distribution for two independent samples, Bull Math Univ Moscou, 2, fasc2 (1939) · Zbl 0023.24902
[48] Castillo, E.; Hadi, A. S.; Balakrishnan, N.; Sarabia, J. M., Extreme value and related models with applications in engineering and science (2005), John Wiley & Sons · Zbl 1072.62045
[50] Benito, B.; Bastida, F., The determinants of the municipal debt policy in Spain, J Public Budget, Account Financ Manage, 16, 4, 525-558 (2004)
[52] Montesinos, V.; Vela, J. M., Governmental accounting in Spain and the european monetary union: a critical perspective, Financ Accountability Manage, 16, 2, 129-150 (2000)
[53] Alba, C.; Navarro, C., Twenty-five years of democratic local government in spain, Reforming local government in Europe, 197-220 (2003)
[54] Solé-Ollé, A., The effects of party competition on budget outcomes: empirical evidence from local governments in Spain, Public Choice, 126, 1-2, 145-176 (2006)
[55] Cabasés, F.; Pascual, P.; Vallés, J., The effectiveness of institutional borrowing restrictions: Empirical evidence from spanish municipalities, Public Choice, 131, 3-4, 293-313 (2007)
[56] Bastida, F.; Benito, B.; Guillamón, M. D., An empirical assessment of the municipal financial situation in Spain, Int Public Manage J, 12, 4, 484-499 (2009)
[57] Hita, F. C.; Orayen, R. E.; Arzoz, P. P., Municipal indebtedness in spain revisited: the impact of borrowing limits and urban development, XVIII Encuentro de economía pública, 9 (2011)
[58] Garcia-Sanchez, I. M.; Prado-Lorenzo, J. M.; Cuadro-Ballesteros, B., Do progressive governments undertake different debt burdens? partisan vs. electoral cycles, Revista de Contabilidad, 14, 1, 29-57 (2011)
[59] Garcia-Sanchez, I. M.; Mordan, N.; Prado-Lorenzo, J., Effect of the political system on local financial condition: empirical evidence for Spain’s largest municipalities, Public Budget Finance, 32, 2, 40-68 (2012)
[60] Lopez-Hernandez, A. M.; Zafra-Gomez, J. L.; Ortiz-Rodriguez, D., Effects of the crisis in spanish municipalities’ financial condition: an empirical evidence (2005-2008), Int J Crit Accoun, 4, 5-6, 631-645 (2012)
[61] Almendral, V. R., The Spanish legal framework for curbing the public debt and the deficit, Eur Constit Law Rev, 9, 02, 189-204 (2013)
[62] Benito, B.; Vicente, C.; Bastida, F., The impact of the housing bubble on the growth of municipal debt: Evidence from Spain, Local Governm Stud, 41, 6, 997-1016 (2015)
[64] Clauset, A.; Shalizi, C. R.; Newman, M. E.J., Power-law distributions in empirical data, SIAM Rev, 51, 4, 661-703 (2009) · Zbl 1176.62001
[65] Hill, B. M., A simple general approach to inference about the tail of a distribution, Ann Stat, 3, 5, 1163-1174 (1975) · Zbl 0323.62033
[66] Brakman, S.; Garretsen, H.; van Marrewikj, C.; van de Berg, M., The return of Zipf: towards a further understanding of the rank-size distribution, J Reg Sci, 39, 1, 182-213 (1999)
[67] Gabaix, X., Zipf’s law and the growth of cities, Am Econ Rev, 89, 2, 129-132 (1999)
[68] Gabaix, X., Zipf’s law for cities: An explanation, Q J Econ, 114, 739-767 (1999) · Zbl 0952.91059
[69] Urzúa, C. M., A simple and efficient test for Zipf’s law, Econ Lett, 66, 257-260 (2000) · Zbl 0951.91059
[70] Ioannides, Y. M.; Overman, H. G., Zipf’s law for cities: an empirical examination, Reg Sci Urban Econ, 33, 127-137 (2003)
[71] Fujiwara, Y.; Guilmi, C. D.; Aoyama, H.; Gallegati, M.; Souma, W., Do Pareto-Zipf and Gibrat laws hold true? an analysis with European firms, Phys A: Stat Mech Appl, 335, 197-216 (2004)
[72] Gabaix, X.; Ioannides, Y. M., The evolution of city size distributions, (Henderson, J. V.; Thisse, J. F., Handbook of regional and urban economics, 4 (2004), Elsevier: Elsevier Amsterdam), 2341-2378
[73] Anderson, H.; Ge, Y., The size distribution of Chinese cities, Reg Sci Urban Econ, 35, 756-776 (2005)
[74] Córdoba, J. C., On the distribution of city sizes, J Urban Econ, 63, 177-197 (2008)
[75] Lomax, K. S., Business failures; another example of the analysis of failure data, J Am Stat Assoc, 49, 847-852 (1954) · Zbl 0056.13702
[76] Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous univariate distributions, vol.1 (1994), John Wiley: John Wiley New York · Zbl 0811.62001
[77] Fisk, P. R., The graduation of income distributions, Econometrica, 29, 171-185 (1961) · Zbl 0104.38702
[78] Burr, I. W., Cumulative frequency functions, Ann Math Stat, 13, 2, 215-232 (1942) · Zbl 0060.29602
[79] Singh, S.; Maddala, G., A function for size distribution of incomes, Econometrica, 44, 5, 963-970 (1976)
[80] Dagum, C., A model of income distribution and the conditions of existence of moments of finite order, Proceedings of the 40th session of the International Statistical Institute, 46, 199-205 (1975) · Zbl 0356.62083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.