Parameter estimation for the truncated Pareto distribution

Journal of the American Statistical Association

Volumn 101, Issue 473, 2006, Pages 270-277

  Aban, Inmaculada B ^a   Meerschaert, Mark M ^b   Panorska, Anna K ^c  

^a UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

^b UNIVERSITY OF OTAGO   (New Zealand)

^c UNIVERSITY OF NEVADA   (United States)

Author keywords

Maximum likelihood estimator; Order statistics; Pareto distribution; Tail behavior; Truncation

Indexed keywords

EID: 33644799765     PISSN: 01621459     EISSN: None     Source Type: Journal    
DOI: 10.1198/016214505000000411     Document Type: Article

References (48)

1. Adler, R., Feldman, R., and Taqqu, M. (1998), A Practical Guide to Heavy Tails, Boston: Birkhäser.
2. Anderson, P., and Meerschaert, M. M. (1997), "Periodic Moving Averages of Random Variables With Regularly Varying Tails," The Annals of Statistics, 25, 771-785.
3. _ (1998), "Modeling River Flows With Heavy Tails," Water Resources Research, 34, 2271-2280.
4. Arnold, B. C. (1983), Pareto Distributions, Burtonsville, MD: International Cooperative Publishing House.
5. Barkai, E., Metzler, R., and Klafter, J. (2000), "From Continuous Time Random Walks to the Fractional Fokker-Planck Equation," Physics Review, Ser. E, 61, 132-138.
6. Beg, M. A. (1981), "Estimation of the Tail Probability of the Truncated Pareto Distribution," Journal of Information & Optimization Sciences, 2, 192-198.
7. _ (1983), "Unbiased Estimators and Tests for Truncation and Scale Parameters," American Journal of Mathematical and Management Sciences, 3, 251-274.
8. Benson, D., Schumer, R., Meerschaert, M. M., and Wheatcraft, S. (2001), "Fractional Dispersion, Levy Motions, and the MADE Tracer Tests," Transport in Porous Media, 42, 211-240.
9. Benson, D., Wheatcraft, S., and Meerschaert, M. M. (2000), "Application of a Fractional Advection-Dispersion Equation," Water Resources Research, 36, 1403-1412.
10. Bickel, P., and Doksum, K. (2001), Mathematical Statistics Basic Ideas and Selected Topics, Vol. 1 (2nd ed.), Upper Saddle River, NJ: Prentice-Hall.
11. Burroughs, S. M., and Tebbens, S. F. (2001a), "Upper-Truncated Power Law Distributions," Fractals, 9, 209-222.
12. _ (2001b), "Upper-Truncated Power Laws in Natural Systems," Journal of Pure and Applied Geophysics, 158, 741-757.
13. _ (2002), "The Upper-Truncated Power Law Applied to Earthquake Cumulative Frequency-Magnitude Distributions," Bulletin of the Seismological Society of America, 92, 2983-2993.
14. Cohen, A. C., and Whitten, B. J. (1988), Parameter Estimation in Reliability and Life Span Models, New York: Marcel Dekker.
15. David, H. (1981), Order Statistics (2nd ed.), New York: Wiley.
16. Douglas, E. M., and Barros, A. P. (2003), "Probable Maximum Precipitation Estimation Using Multifractals: Application in the Eastern United States," Journal of Hydrometeorology, 4, 1012-1024.
17. Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997), Modelling Extremal Events for Insurance and Finance, Berlin: Springer-Verlag.
18. Fama, E. (1965), "The Behavior of Stock Market Prices," Journal of Business, 38, 34-105.
19. Feller, W. (1971), An Introduction to Probability Theory and Its Applications, Vol. II (2nd ed.), New York: Wiley.
20. Fofack, H., and Nolan, J. P. (1999), 'Tail Behavior, Modes and Other Characteristics of Stable Distribution," Extremes, 2, 39-58.
21. Groisman, P. Y, Knight, R. W., Karl, T. R., Easterling, D. R., Sun, B., and Lawrimore, J. H. (2004), "Contemporary Changes of the Hydrological Cycle Over the Contiguous United States: Trends Derived From in Situ Observations," Journal of Hydrometeorology, 5, 64-85.
22. Hall, P. (1982), "On Some Simple Estimates of an Exponent of Regular Variation," Journal of the Royal Statistical Society, Ser. B, 44, 37-42.
23. Hill, B. (1975), "A Simple General Approach to Inference About the Tail of a Distribution," The Annals of Statistics, 3, 1163-1173.
24. Hosking, J., and Wallis, J. (1987), "Parameter and Quantile Estimation for the Generalized Pareto Distribution," Technometrics, 29, 339-349.
25. Jansen, D., and de Vries, C. (1991), "On the Frequency of Large Stock Market Returns: Putting Booms and Busts Into Perspective," Review of Economics and Statistics, 73, 18-24.
26. Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994), Continuous Univariate Distributions (2nd ed.), New York: Wiley.
27. Klafter, J., Blumen, A., and Shlesinger, M. F. (1987), "Stochastic Pathways to Anomalous Diffusion," Physics Review, Ser. A, 35, 3081-3085.
28. Kotulski, M. (1995), "Asymptotic Distributions of the Continuous Time Random Walks: A Probabilistic Approach," Journal of Statistic Physics, 81, 777-792.
29. Loretan, M., and Phillips, P. (1994), "Testing the Covariance Stationarity of Heavy-Tailed Time Series," Journal of Empirical Finance, 1, 211-248.
30. Lu, S. L., and Molz, F. J. (2001), "How Well Are Hydraulic Conductivity Variations Approximated by Additive Stable Processes?" Advances in Environmental Research, 5, 39-45.
31. Malamud, B. D., Morein, G., and Turcotte, D. L. (1998), "Forest Fires: An Example of Self-Organized Critical Behavior," Science, 281, 1840-1842.
32. Mandelbrot, B. (1963), "The Variation of Certain Speculative Prices," Journal of Business, 36, 394-419.
33. McCulloch, J. (1996), "Financial Applications of Stable Distributions," in Statistical Methods in Finance, eds. G. Madfala and C. R. Rao, Amsterdam: Elsevier, pp. 393-425.
34. Meerschaert, M. M., Benson, D. A., Scheffler, H. P., and Becker-Kern, P. (2002), "Governing Equations and Solutions of Anomalous Random Walk Limits," Physics Review, Ser. E, 66, 102-105.
35. Meerschaert, M. M., and Scheffler, H. P. (2003), "Portfolio Modeling With Heavy-Tailed Random Vectors," in Handbook of Heavy-Tailed Distributions in Finance, ed. S. T. Rachev, New York: Elsevier, pp. 595-640.
36. Metzler, R., and Klafter, J. (2000), "The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach," Physics Report, 339, 1-77.
37. Nikias, C., and Shao, M. (1995), Signal Processing With Alpha-Stable Distributions and Applications, New York: Wiley.
38. Pacheco, J., Scholtz, C. H., and Sykes, L. R. (1992), "Changes in Frequency-Size Relationship From Small to Large Earthquakes," Nature, 355, 71-73.
39. Rachev, S., and Mittnik, S. (2000), Stable Paretian Models in Finance, Chichester, U.K.: Wiley.
40. Rehfeldt, K. R., Boggs, J. M., and Gelhar, L. W. (1992), "Field Study of Dispersion in a Heterogeneous Aquifer. 3: Geostatistical Analysis of Hydraulic Conductivity," Water Resources Research, 28, 3309-3324.
41. Resnick, S. (1997), "Heavy Tail Modeling and Teletraffic Data," The Annals of Statistics, 25, 1805-1869.
42. Resnick, S., and Stǎricǎ, C. (1995), "Consistency of Hill's Estimator for Dependent Data," Journal of Applied Probability, 32, 139-167.
43. Samorodnitsky, G., and Taqqu, M. (1994), Stable Non-Gaussian Random Processes, New York: Chapman & Hall.
44. Scholz, C. H., and Contreras, J. C. (1998), "Mechanics of Continental Rift Architecture," Geology, 26, 967-970.
45. Schumer, R., Benson, D., Meerschaert, M. M., and Wheatcraft, S. (2001), "Eulerian Derivation of the Fractional Advection-Dispersion Equation," Journal of Contaminant Hydrology, 38, 69-88.
46. Thompson, C. S., and Tomlinson, A. I. (1995), "A Guide to Probable Maximum Precipitation in New Zealand," Science and Technolody Series Technical Note ST19 (http://www.niwa.co.nz/pubs/st), National Institute of Water & Atmospheric Research of New Zealand.
47. Uchaikin, V., and Zolotarev, Z. (1999), Chance and Stability: Stable Distributions and Their Applications, Utrecht: VSP.
48. World Meteorological Organization (1986), Manual for Estimation of Probable Maximum Precipitation, Geneva: Author.