2,291
Views
3
CrossRef citations to date
0
Altmetric
Articles

Estimation of the Pareto and related distributions – A reference-intrinsic approach

& ORCID Icon
Pages 523-542 | Received 27 Nov 2019, Accepted 07 Apr 2021, Published online: 11 May 2021

References

  • Akhundjanov, S. B., and L. Chamberlain. 2019. The power-law distribution of agricultural land size. Journal of Applied Statistics 46 (16):3044–56. doi:10.1080/02664763.2019.1624695.
  • Behrens, C. N., H. F. Lopes, and G. Gamerman. 2004. Bayesian analysis of extreme events with threshold estimation. Statistical Modelling 4 (3):227–44. doi:10.1191/1471082X04st075oa.
  • Beirlant, J., Y. Goegebeur, J. Teugels, and J. Segers. 2005. Statistics of Extremes: Theory and Applications. Chichester: Springer.
  • Beirlant, J., J. Teugels, and P. Vynckier. 1996. Practical analysis of extreme values. Leuven: University Press.
  • Berger, J., J. M. Bernardo, and D. Sun. 2009. The formal definition of reference priors. The Annals of Statistics 37 (2):905–38. doi:10.1214/07-AOS587.
  • Berger, J., J. M. Bernardo, and D. Sun. 2015. Overall objective priors. Bayesian Analysis 10 (1):189–221. doi:10.1214/14-BA915.
  • Bernardo, J. M. 2007. Objective Bayesian point and region estimation in location-scale models. Statistics and Operations Research Transactions 31:3–44.
  • Bernardo, J. M., and M. A. Juárez. 2003. Intrinsic estimation. Bayesian Statistics 7, eds. J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, and M. West, 465–76. Oxford: Oxford University Press.
  • Bernardo, J. M., and R. Rueda. 2002. Bayesian hypothesis testing: A reference approach. International Statistical Review 70 (3):351–72. doi:10.1111/j.1751-5823.2002.tb00175.x.
  • Castellanos, M. E., and S. Cabras. 2007. A default Bayesian procedure for the generalized Pareto distribution. Journal of Statistical Planning and Inference 137 (2):473–83. doi:10.1016/j.jspi.2006.01.006.
  • Castillo, E., and A. S. Hadi. 1997. Fitting the generalized Pareto distribution to data. Journal of the American Statistical Association 92 (440):1609–20. doi:10.1080/01621459.1997.10473683.
  • Castillo, E., A. S. Hadi, N. Balakrishnan, and J. M. Sarabia. 2004. Extreme value and related models with applications in engineering and science. New Jersey: Wiley.
  • Charras-Garrido, M., and P. Lezaud. 2013. Extreme value analysis: An introduction. Journal de la Société Française de Statistique 154:66–97.
  • Chen, H., W. Cheng, J. Zhao, and X. Zhao. 2017. Parameter estimation for generalized Pareto distribution by generalized probability weighted moment-equations. Communications in Statistics - Simulation and Computation 46 (10):7761–76. doi:10.1080/03610918.2016.1249884.
  • Coles, S. 2001. An introduction to statistical modeling of extreme values. London: Springer-Verlag.
  • Coles, S., L. R. Pericchi, and S. Sisson. 2003. A fully probabilistic approach to extreme rainfall modeling. Journal of Hydrology 273 (1–4):35–50. doi:10.1016/S0022-1694(02)00353-0.
  • Davidson, A. C., and R. L. Smith. 1990. Models for exceedances over high thresholds. Journal of the Royal Statistical Society B 52:393–442.
  • Diebolt, J., A. Guillou, and R. Worms. 2003. Asymptotic behaviour of the probability weighted moments and penultimate approximation. ESAIM: Probability and Statistics 7:219–38. doi:10.1051/ps:2003010.
  • Diebolt, J., M. A. El-Aroui, M. Garrido, and S. Girard. 2005. Quasi-conjugate Bayes estimates for GPD parameters and application to heavy tails modelling. Extremes 8 (1–2):57–78. doi:10.1007/s10687-005-4860-9.
  • Dupuis, D., and M. Tsao. 1998. A hybrid estimator for generalized Pareto and extreme-value distributions. Communications in Statistics - Theory and Methods 27 (4):925–41. doi:10.1080/03610929808832136.
  • Embrechts, P., C. Klüppelberg, and T. Mikosch. 1997. Modelling extremal events. Berlin: Springer-Verlag.
  • Engeland, K., H. Hisdal, and A. Frigessi. 2004. Practical extreme value modelling of hydrological floods and droughts: A case study. Extremes 7 (1):5–30. doi:10.1007/s10687-004-4727-5.
  • Fawcett, L., and D. Walshaw. 2006. A hierarchical model for extreme wind speeds. Journal of the Royal Statistical Society: Series C (Applied Statistics) 55 (5):631–46. doi:10.1111/j.1467-9876.2006.00557.x.
  • Fletcher, R., and C. M. Reeves. 1964. Function minimization by conjugate gradients. The Computer Journal 7 (2):149–54. doi:10.1093/comjnl/7.2.149.
  • Frigessi, A., O. Haug, and H. Rue. 2002. A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes 5 (3):219–35. doi:10.1023/A:1024072610684.
  • Galambos, J. 1987. The asymptotic theory of extreme order statistics, 2nd edn. New York: Wiley.
  • Ghosh, S., and S. Resnick. 2010. A discussion on mean excess plots. Stochastic Processes and Their Applications 120 (8):1492–517. doi:10.1016/j.spa.2010.04.002.
  • Gilleland, E., M. Ribatet, and A. C. Stephenson. 2013. A software review for extreme value analysis. Extremes 16 (1):103–19. doi:10.1007/s10687-012-0155-0.
  • Goldstein, M. L., S. A. Morris, and G. G. Yen. 2004. Problems with fitting to the power-law distribution. The European Physical Journal B 41 (2):255–8. doi:10.1140/epjb/e2004-00316-5.
  • Greenwood, J. A., J. M. Landwehr, N. C. Matalas, and J. R. Wallis. 1979. Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form. Water Resources Research 15 (5):1049–54. doi:10.1029/WR015i005p01049.
  • de Haan, L., and A. F. Ferreira. 2006. Extreme value theory: An introduction. New York: Springer.
  • Hill, B. M. 1975. A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3 (5):1163–74. doi:10.1214/aos/1176343247.
  • Holmes, J., and W. Moriarty. 1999. Application of the generalized Pareto distribution to extreme value analysis in wind engineering. Journal of Wind Engineering and Industrial Aerodynamics 83 (1–3):1–10. doi:10.1016/S0167-6105(99)00056-2.
  • Hosking, J. R. M., J. R. Wallis, and E. F. Wood. 1985. Estimation of the generalized extreme value distribution by the method of probability weighted moments. Technometrics 27 (3):251–61. doi:10.1080/00401706.1985.10488049.
  • Jagger, T. H., and J. B. Elsner. 2006. Climatology models for extreme hurricane winds near the United States. Journal of Climate 19 (13):3220–36. doi:10.1175/JCLI3913.1.
  • Juárez, S. F., and W. R. Schucany. 2004. Robust and efficient estimation for the generalized Pareto distribution. Extremes 7 (3):237–51. doi:10.1007/s10687-005-6475-6.
  • Juárez, M. A. 2005. Objective Bayes estimation and hypothesis testing: The reference-intrinsic approach on non-regular models. CRISM Working Paper 05–14.
  • Keylock, C. 2005. An alternative form for the statistical distribution of extreme avalanche runout distances. Cold Regions Science and Technology 42 (3):185–93. doi:10.1016/j.coldregions.2005.01.004.
  • Koch, R. 2007. The 80/20 principle: The secret of achieving more with less. London: Nicholas Brealey Publishing.
  • Kotz, S., and S. Nadarajah. 2000. Extreme value distributions: Theory and applications. London: Imperial College Press.
  • Krehbiel, T., and L. C. Adkins. 2008. Extreme daily changes in U.S. dollar London inter-bank offer rates. International Review of Economics & Finance 17 (3):397–411. doi:10.1016/j.iref.2006.08.009.
  • La Cour, B. R. 2004. Statistical characterization of active sonar reverberation using extreme value theory. IEEE Journal of Oceanic Engineering 29 (2):310–6. doi:10.1109/JOE.2004.826897.
  • Lana, X., A. Burgueño, M. Martínez, and C. Serra. 2006. Statistical distributions and sampling strategies for the analysis of extreme dry spells in Catalonia (NE Spain). Journal of Hydrology 324 (1–4):94–114. doi:10.1016/j.jhydrol.2005.09.013.
  • Leadbetter, M. R., G. Lindgren, and H. Rootzen. 1983. Extremes and related properties of random sequences and processes. New York: Springer.
  • Lima, C. H. R., U. Lall, T. Troy, and N. Devineni. 2016. A hierarchical Bayesian GEV model for improving local and regional flood quantile estimates. Journal of Hydrology 541:816–23. doi:10.1016/j.jhydrol.2016.07.042.
  • Malik, H. J. 1970. Estimation of the parameters of a Pareto distribution. Metrika 15 (1):126–32. doi:10.1007/BF02613565.
  • McNeil, A. J., R. Frey, and P. Embrechts. 2005. Quantitative risk management: Concepts, techniques and tools. Princeton: University Press.
  • McNeil, A. J. 1997. Estimating the tails of loss severity distributions using extreme value theory. ASTIN Bulletin 27 (1):117–37. doi:10.2143/AST.27.1.563210.
  • Mendes, J. M., P. C. de Zea Bermudez, J. Pereira, K. F. Turkman, and M. J. P. Vasconcelos. 2010. Spatial extremes of wildfire sizes: Bayesian hierarchical models for extremes. Environmental and Ecological Statistics 17 (1):1–28. doi:10.1007/s10651-008-0099-3.
  • Moharram, S. H., A. K. Gosain, and P. N. Kapoor. 1993. A comparative study for the estimators of the generalized Pareto distribution. Journal of Hydrology 150 (1):169–85. doi:10.1016/0022-1694(93)90160-B.
  • Moisello, U. 2007. On the use of partial probability weighted moments in the analysis of hydrological extremes. Hydrological Processes 21 (10):1265–79. doi:10.1002/hyp.6310.
  • Öztekin, T. 2005. Comparison of parameter estimation methods for the three-parameter generalized Pareto distribution. Turkish Journal of Agriculture and Forestry 29:419–28.
  • Pandey, M. D., P. H. A. J. M. Van Gelder, and J. K. Vrijling. 2001. The estimation of extreme quantiles of wind velocity using L-moments in the peaks-over-threshold approach. Structural Safety 23 (2):179–92. doi:10.1016/S0167-4730(01)00012-1.
  • Pandey, M. D., P. H. A. J. M. Van Gelder, and J. K. Vrijling. 2004. Dutch case studies of the estimation of extreme quantiles and associated uncertainty by bootstrap simulations. Environmetrics 15 (7):687–99. doi:10.1002/env.656.
  • Peng, L., and A. Welsh. 2001. Robust estimation of the generalized Pareto distribution. Extremes 4 (1):53–65. doi:10.1023/A:1012233423407.
  • Persky, J. 1992. Retrospectives: Pareto’s law. Journal of Economic Perspectives 6 (2):181–92. doi:10.1257/jep.6.2.181.
  • Pickands, J. 1975. Statistical inference using extreme order statistics. The Annals of Statistics 3:119–31.
  • Pisarenko, V. F., and D. Sornette. 2003. Characterization of the frequency of extreme earthquake events by the generalized Pareto distribution. Pure and Applied Geophysics 160 (12):2343–64. doi:10.1007/s00024-003-2397-x.
  • R Core Team. 2021. R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. https://www.R-project.org/
  • Ragulina, G., and T. Reitan. 2017. Generalized extreme value shape parameter and its nature for extreme precipitation using long time series and the Bayesian approach. Hydrological Sciences Journal 62 (6):863–79. doi:10.1080/02626667.2016.1260134.
  • Robert, C. P. 1996. Intrinsic losses. Theory and Decision 40 (2):191–214. doi:10.1007/BF00133173.
  • Rootzén, H., and N. Tajvidi. 1997. Extreme value statistics and wind storm losses: A case study. Scandinavian Actuarial Journal 1997 (1):70–94. doi:10.1080/03461238.1997.10413979.
  • Rytgaard, M. 1990. Estimation in the Pareto distribution. ASTIN Bulletin 20 (2):201–16. doi:10.2143/AST.20.2.2005443.
  • Shi, G., H. Atkinson, C. Sellars, and C. Anderson. 1999. Application of the generalized Pareto distribution to the estimation of the size of the maximum inclusion in clean steels. Acta Materialia 47 (5):1455–68. doi:10.1016/S1359-6454(99)00034-8.
  • Tancredi, A., C. Anderson, and A. O’Hagan. 2006. Accounting for threshold uncertainty in extreme value estimation. Extremes 9 (2):87–106. doi:10.1007/s10687-006-0009-8.
  • Vilar-Zanón, J. L., and C. Lozano-Colomer. 2007. On Pareto conjugate priors and their application to large claims reinsurance premium calculation. ASTIN Bulletin 37 (2):405–28. doi:10.1017/S0515036100014938.
  • White, E. P., B. J. Enquist, and J. L. Green. 2008. On estimating the exponent of power-law frequency distributions. Ecology 89:905–12. doi:10.1890/07-1288.1.
  • Zagorski, M., and M. Wnek. 2007. Analysis of the turbine steady-state data by means of generalized Pareto distribution. Mechanical Systems and Signal Processing 21 (6):2546–59. doi:10.1016/j.ymssp.2007.01.002.
  • de Zea Bermudez, P., and S. Kotz. 2010a. Parameter estimation of the generalized Pareto distribution–Part I. Journal of Statistical Planning and Inference 140 (6):1353–73. doi:10.1016/j.jspi.2008.11.019.
  • de Zea Bermudez, P., and S. Kotz. 2010b. Parameter estimation of the generalized Pareto distribution–Part II. Journal of Statistical Planning and Inference 140 (6):1374–88. doi:10.1016/j.jspi.2008.11.020.
  • de Zea Bermudez, P., J. Mendes, J. M. C. Pereira, K. F. Turkman, and M. J. P. Vasconcelos. 2009. Spatial and temporal extremes of wildfire sizes in Portugal (1984–2004). International Journal of Wildland Fire 18 (8):983–91. doi:10.1071/WF07044.
  • de Zea Bermudez, P., and M. A. A. Turkman. 2003. Bayesian approach to parameter estimation of the generalized Pareto distribution. Test 12 (1):259–77. doi:10.1007/BF02595822.